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THE THIRTEEN BOOKS 

OF 

EUCLID'S ELEMENTS 

TRANSLATED FROM THE TEXT OF HEIBERG 

WITH INTRODUCTION AND COMMENTARY 

BY 
T. L. HEATH, C.B., Sc.D., 

SOMBTIMK FELLOW OF TRINmr COLLKGB, CAMBMDGB 



VOLUME I 
INTRODUCTION AND BOOKS 1, H 



* • . • 



Cambridge : 
at the University Press 

1908 

1*1 



I 



THE THIRTEEN BOOKS 

•OF 

EUCLID'S ELEMENTS 



CAMBRIDGE UNIVERSITY PRESS WAREHOUSE 

C. F. CLAY, Manager. 

Eiltei: FETTER LANE, E.C. 

•Mrtml: too. PRINCES STREET. 




Ikrlte: A. ASHSR AND 00. 
; P. A. BROCKHAUS, 
G. P. PUTNAM'S SON& 
n» Calcvlte: IfACMILLAN AND CO., Lm 



[Al/ X^Jk/s merved.] 



A***r 



XT rT»« 



•K(^i«4P4« 'tjV*^ -HmP V^ fttC J? . ^ l^if. iVTi fi- uj 







Chap. 


I. 


. »» 


II. 


»> 


III. 


>» 


IV. 


»» 


V. 


»« 


VI. 


M 


VII. 


t *> 


VIII. 


5 

f " 


IX. 



CONTENTS 

VOLUME I. 

INTRODUCTION. 

p. 

eucud and the traditions about him 

Euclid's other works 

Greek commentators other than Proclus 

Proclus and his sources 

The Text 

The Scholia 

Euclid in Arabia 

PRiNaPAL translations and editions . 

§ I. On the nature op Elements i 

§ 2. Elements anterior to Euclid's . . . : 
I §3. First principles: Depinitiqns, Postulates 

AND Axioms . . . ' . i 

[ § 4. Theorems and Problems . . . . j 

( § 5. The formal divisions of a proposition i 

h § 6. Other technical terms . . . . : 

§ 7. The definitions i 

[ THE ELEMENTS. 

I Book I. Definitions, Postulates, Common Notions . ] 

Notes on Definitions etc. ] 

Propositions . j 

Book II. Definitions 

Note on geometrical algebra 
Propositions . ^ . . . 





— 








•— 






k 


X 




CONTENTS 
VOLUME II. 


Book 


IIL 


Definitions . 






Propositions , 




Book 


IV, 


Definitions 
Propositions , 




Book 


V. 


Introductory hote 
Definitjons , 

Propositions , 




Book 


VL 


Introductory hote 

Definitions . 
Propositions . 




Book 


VIL 


Definitions 

Propositions , 




Book 


VIIL 


• 1 




Book ;IX, 


* 




Greek Index to Vol, H. . 




FjfGLiSH Ihdex. to Vol. 1L 






FAOt 



t 

?8 

119 

Its 

138 

18S 
191 

«77 
99$ 

345 
384 

431 



VOLUME IlL 

Book X. Introductory note i 

Definitions 10 

Propositions i^ — ^47 . , . , • ,14-101 

Definitions il ,/,.... ioi 

Propositions 4S — 84 iq?-i77 

Definitions hi ryj 

Propositions 85—115 17&-254 

Ancient extensions of theory of Book X . 255 

Book XL Definitions . * a6o 

Propositions . - - ' 27a 

Book XIL Historical note . . < . ' • . . 365 

Propositions 369 

Book XIII* Historical note 43S 

Propositions ......*. 44a 

iPPENDix. L The so-called "Book XI V," (by Hypsicles) 511 

IL Note on the so-called **Book XV/* , , 519 

iddknda et corrigenda .*...... £21 

* Index: Greek 329 

English 535 



] 



n 



ifir Christoph Schlussel rtad Christoph KUn 



ERRATA 

Vol. I. p. i9» line \*i^fcr ** bat not a platfonn and sixpence " rtid " bat not a figare and 
tizpence" 

» P> io5» luM lO 

»• p* 415* col* <• Une 17 
Vol. II. p. 41, line 97 
Vol. III. p. 539, col. I, line 45 
VoL I. p. 106, line i \ 

99 P* 415* col. I, line ai y/or Giacomo read Giovanni 
VoL III. p. 558, coL a, line 95 j 
VoL I. p. 18a, line %\^ for Opot read^Opn 
Vol. III. p. 499. In the 6gnre» CV ihoald be a thick line. 

iTM'l EUCUD 



PREFACE 

\"nnHERE never has been, and till we see it we nev< 
^ X shall believe that there can be, a system of geometi 
worthy of the name, which has any material departures (we d 
not speak of corrections or extensions or developments) froi 
the plan laid down by Euclid," De Morgan wrote thus i 
October 1848 (Short supplementary remarks on the first s\ 
Books of EuclicTs Elements in the Companion to the Almam 
for 1849); and I do not think that, if he had been livin 
to-day, he would have seen reason to revise the opinion s 
deliberately pronounced sixty years ago. It is true that in th 
interval much valuable work has been done on the continei 
in the investigation of the first principles, including th 

. formulation and classification of axioms or postulates whic 
are necessary to make good the deficiencies of Euclid's ow 
explicit postulates and axioms and to justify the furth< 
assumptions which he tacitly makes in certain proposition 
content apparently to let their truth be inferred from observ; 
tion of the figures as drawn ; but, once the first principles ai 
disposed of, the body of doctrine contained in the recent tex 
books of elementary geometry does not, and from the natui 
of the case cannot, show any substantial differences from th; 
set forth in the Elements. In England it would seem that f< 
less of scientific value has been done ; the efforts of a multitud 
of writers have rather been directed towards producing alte; 
natives for Euclid which shall be more suitable, that is to sa; 
easier, for schoolboys. It is of course not surprising that, i 



vi PREFACE 

these days of short cuts, there should have arisen a movement 
to get rid of Euclid and to substitute a *' royal road to 
geometry " ; the marvel is that a book v^^hich was not written 
for schoolboys but for grown men (as all internal evidence 
shows^ and in particular the essentially theoretical character 
of the work and its aloofness from anything of the nature of 
" practical " geometry) should have held its own as a school- 
book for so long» And now that Euclid's proofs and arrange- 
ment are no longer required from candidates at examinations 
there has been a rush of competitors anxious to be first in the 
field with a new text-book on the more " practical " lines which 
now find so much favour. The natural desire of each teacher 
who writes such a text- book is to give prominence to some 
special nostrum which he has found successful with pupils. 
One result is, too often, a loss of a due sense of proportion ; 
and, in any case, it is inevitable that there should be great 
diversity of treatment. It was with reference to such a danger 
that Lardner wrote in 1846 : '* Euclid once superseded, every 
teacher would esteem his own work the best, and every school 
would have its own class book. All that rigour and exactitude 
which have so long excited the admiration of men of science 
would be at an end. These very words would lose all definite 
meaning. Every school would have a different standard ; 
matter of assumption in one being matter of demonstration in 
another; until, at length, Geometry, in the ancient sense of 
the word, would be altogether frittered away or be only 
considergd as a particular application of Arithmetic and 
Algebra/j It is, perhaps, too early yet to prophesy what will 
be the ultimate outcome of the new order of things ; but it 
would at least seem possible that history will repeat itself and 
that, when chaos has come again in geometrical teaching, 
there will be a return to Euclid more or less complete for the 
purpose of standardising it once more. 

But the case for a new edition of Euclid is independent of 
any controversies as to how geometry shall be taught to 
schoolboys. Euclid's work will live long after all the text-books 



PREFACE 



(of the present day are superseded and forgotten. It is on 
of the noblest monuments of antiquity ; no mathematicia 
worthy of the name can afford not to know Euclid, the r« 
Euclid as distinct from any revised or rewritten version 
which will serve for schoolboys or engineers. And, to kno^ 
L Euclid, it is necessary to know his language, and, so far as 
' can be traced, the history of the "elements" which h 
; collected in his immortal work. 

\ This brings me to the raison d'Hre of the present editioi 

A new translation from the Greek was necessary for tw 
I reasons. First, though some time has elapsed since th 
'^ appearance of Heiberg's definitive text and prolegomens 
I published between 1883 and 1888, there has not been, so fa 
^ as I know, any attempt to make a faithful translation from 
I into English even of the Books which are commonly reac 
^ And, secondly, the other Books, vii. to x. and xiii., were nc 
' included by Simson and the editors who followed him, c 
: apparently in any English translation since Williamson 
(178 1 — 8), so that they are now practically inaccessible t 
English readers in any form. 

In the matter of notes, the edition of the first six Book 

' in Greek and Latin with notes by Camerer and Haubc 

(Berlin, 1824 — 5) is a perfect mine of information. It woul 

h have been practically impossible to make the notes mor 

\ exhaustive at the time when they were written. But th 

researches of the last thirty or forty years into the history < 

mathematics (I need only mention such names as those < 

r Bretschneider, Hankel, Moritz Cantor, Hultsch, Paul Tannery 

j^ Zeuthen, Loria, and Heiberg) have put the whole subjec 

i( upon a different plane. I have endeavoured in this editio 

to take account of all the main results of these researches u 

to the present date. Thus, so far as the geometrical Book 

^ are concerned, my notes are intended to form a sort c 

(f dictionary of the history of elementary geometry, arrange 

according to subjects ; while the notes on the arithmetic* 

Books VII. — IX. and on Book x. follow the same plan. 



viii PREFACE 

I desire to express here my thanks to my brother, 
Dr R. S* Heath, Vice- Principal of Birmingham University, 
for suggestions on the proof sheets and, in particular p for the 
reference to the parallelism between Euclid's definition of 
proportion and Dedekind's theory of irrationals, to Mr R. D* 
Hicks for advice on a number of difficult points of translation, 
to Professor A. A. Bevan for help in the transliteration of 
Arabic names, and to the Curators and Librarian of the 
Bodleian Library for permission to reproduce, as frontispiece, 
a page from the famous Bodleian MS. of the Ehments^ 
Lastly, my best acknowledgments are due to the Syndics of 
the Cambridge University Press for their ready acceptance 
of the work, and for the zealous and efficient cooperation of 
their staff which has much lightened the labour of seeing the 
book through the Press, 

T. U H, 
NovembcTy 1908. 



X 



• • • 



r . ■.•••: 

• • • 



^ INTRODUCTION. 

I 



CHAPTER I. 

EUCLID AND THE TRADITIONS ABOUT HIM. 

As in the case of the other great mathematicians of Greece, so in 
Euclid's case, we have only the most meagre particulars of the life 
and personality of the man^ 

Most of what we have is contained in the passage of Proclus' 
summary relating to him, which is as follows^: 

" Not much younger than these (sc. Hermotimus of Colophon and 
Philippus of Mende) is Euclid, who put together the Elements, collect- 
ing many of Eudoxus' theorems, perfecting many of Theaetetus', and 
also bringing to irrefragable demonstration the things which were 
only somewhat loosely proved by his predecessors. This man lived* 
in Uie time of the first Ptolemy. For Archimedes, who came imme- 
diately after the first (Ptolemy)', makes mention of Euclid: and, 
further, they say that Ptolemy once asked him if there was in 
geometry any shorter way than that of the elements, and he answered 
that there was no royal road to geometry*. He is then younger than 
the pupils of Plato but older than Eratosthenes and Archimedes ; for 
the latter were contemporary with one another, as Eratosthenes some- 
where says." 

This passage shows that even Proclus had no direct knowledge 
^ of Euclici s birthplace or of the date of his birth or death. He pro- 
- ceeds by inference. Since Archimedes lived just after the first 

I ^ Proclus, ed. Friedlein, p. 68, 6 — lo. 

' * The word yfywe must apparently mean ''flourished," as Heiberg understands it 

^ ^LitterargeschichtNche StudUn iiber Euklid, 1881, p. 16), not "was bom," as Hankel took 

it : otherwise part of Proclus' argument would lose its cogency. 
^ 'So Heiberg understands /iri/SoXiiw r^ wptimfi (sc. nroXefuUy). Friedlein's text has 

I Kol between iwifioKunf and nf wptirtfi; and it is rignt to remark that another reading is 
U KolipTif wpilbrtf (without irtfiaXtitf) which has been translated " in his first ^>(," by wmch 
is understood On the Sphere and Cylmder i., where (i) in Prop- ^ ure the words *' let BC 
be made equal to D by the seamd (proposition) of the first of Euclid's (books)," and (a) in 
Prop. 6 the words " For these things are handed down in the Elements " (without the name 
of Euclid). Heiberg thinks the former passage is rdPerred to, and tnat Proclus must 
therefore have had before him the words " by the second of the first of Euclid ** : a fair proof 
that they are genuine, though in themselves &iey would be somewhat suspicious, 
f ^ The same story is told in Stobaeus, Eel, (11. p. 338, 30, ed. Wachsmuth) about 

Alexander and Menaechmus. Alexander is represented as having asked Menaechmiis to 
teach him geometry concisely, but he replied : '*0 kinff, through the country there are royal 
roads and roads for common citixens, but in geometry there is one road for all.*' 

H. E. "^ I 



t 



f 



y 



4 



t u/'", INTRODUCTION [gh j 

Ptolemy/'ftoS Archimedes mentions Euclid^ while there is an anecdote 
dhout'J^f^e Ptolemy and Euclid, therefore EucUd hVed in the time of 
the fii^l Ptolemy. 

.W^'may infer then from Proclus that Euclid was intermediate 

between the first pupils of Plato and Archimedes. Now Plato died in 

■3^ J Archimedes lived 287-212, Eratosthenes 276-194 B.C Thus 

r, Ej/clid must have flourished c. 300 B,C, which date agrees well with 

V*the fact that Ptolemy reigned from 306 to 2 S3 B.C 

/•■ It is most probable that Euclid received his mathematical training 

* in Athens from the pupils of Plato; for most of the geometers who 
could have taught him were of that school, and it was in Athens that 
the older writers of elements, and the other mathematicians on whose 
works Euclid's Elemtnts depend, had lived and taught He may 
himself have been a Platonist, but this does not follow from the state- 
ments of Proclus on the subject. Proclus says namely that he was of 
the school of Plato and in close touch with that philosophy \ But 
this was only an attempt of a New Platonist to connect Euclid with 
his philosophy I as is clear from the next words in the same sentence, 
•* for which reason also he set before himself, as the end of the whole 
Elements, the construction of the so-called Platonic figures" It is 
evident that it was only an idea of Proclus* own to infer that Euclid 
was a Platonist because his Elemmts end with the investigation of 
the five regular solids, since a later passage shows him hard put to 
it to reconcile the view that the construction of the five regular solids 
was the end and aim of the Eiements with the obvious fact that they 
were intended to supply a foundation for the study of geometry in 
general, "to make perfect the understanding of the learner in regard - 
to the whole of geometry'" To get out of the difficulty he says' that, 
if one should ask him what was the aim (<rico7ro9) of the treatise, he 
would reply by making a distinction between Euclid's intentions 
(i) as regards the subjects with which his investigations are concerned, 
(2) as regards the learner, and would say as regards (I) that "the 
whole of the geometer's argument is concerned with the cosmic 
figures." This latter statement is obviously incorrect It is true 
that Euclid's Elements end with the construction of the five regular 
solids; but the plan i metrical portion has no direct relation to them, 
and the arithmetical no relation at all ; the propositions about them 
are merely the conclusion of the stereo metrical division of the work. 

One thing is however certain, namely that Euclid taught, and 
founded a school, at Alexandria, This is clear from the remark of 
Pappus about Apollonius*: *'he spent a very long time with the 
pupils of Euclid at Alexandria, and it was thus that he acquired 
such a scientific habit of thought** 

It is in the same passage that Pappus makes a remark which 
might J to an unwary reader, seem to throw some light on the 

^ Proclus, p. ^1 90, Kol rf rpoatp^ct £j nXorbuvtr&f iart jral rj ^X«<ra^9 Tm:CfrQ alxiUitt. 

■ (W^, p* 71, 8. * ibid. p» 70, 19 ^qq, 

* PttppU^, VII. p* ISjSt 10— iIt inAT^oXd^at r«r itwh £l^j(X«iJcM^ pud^itmlt ir 'AXtfardpeffi 



I 



CH. i] EUCLID AND TRADITIONS ABOUT HIM 

personality of Euclid. He is speaking about Apollonius' prefac 

to the first book of his Conies^ where he says that Euclid had no 

I completely worked out the synthesis of the "three- and four-Iin< 

r* locus," which in fact was not possible without some theorems firs 

discovered by himself. Pappus says on this^: "Now Euclid- 

"^ regarding Aristaeus as deserving credit for the discoveries he ha( 

already made in conies, and without anticipating him or wishing t< 

construct ane\v the same system (such was his scrupulous fairness an( 

V his exemplary kindliness towards all who could advance mathematica 

' science to however small an extent), being moreover in no wise con 

tentious and, though exact, yet no braggart like the other [ApoUonius 

. — ^wrote so much about the locus as was possible by means of th< 

• conies of Aristaeus, without claiming completeness for his demonstra 

tions." It is however evident, when the passage is examined in it 

context, that Pappus is not following any tradition in giving thi 

^ account of Euclid: he was offended by the terms of Apolloniuj 

' reference to Euclid, which seemed to him unjust, and he drew \ 

\ fancy picture of Euclid in order to show ApoUonius in a relativel] 

^ unfavourable light 
Another story is told of Euclid which one would like to believe true 

I According to^tobaeus*, " some one who had begun to read geometr 
with EuclidT^en he had learnt the first theorem, asked Euclid, ' Bu 

f what shall 1 get by learning these things ? ' Euclid called his slav 

\ and said ' Give him threepence, since he must make gain out of wha 

j he learns.'" 

i In the middle ages most translators and editors spoke of Eucli( 

as Euclid of Megara, This description arose out of a confusioi 
between our Euclid and the philosopher Euclid of Megara who livec 
about 400 B.C The first trace of this confusion appears in Valeriu 
Maximus (in the time of Tiberius) who says' that Plato, on beinj 
appealed to for a solution of the problem of doubling the cubica 
altar, sent the inquirers to "Euclid the geometer." There is no doub 
about the reading, although an early commentator on Valeriu 
Maximus wanted to correct "Eucliden" into **EudoxufnP and thi 
correction is clearly right. But, if Valerius Maximus took Euclid th( 
geometer for a contemporary of Plato, it could only be througl 
confusing him with Euclid of Megara. The first specific reference t 
Euclid as Euclid of Megara belongs to the 14th century, occurring i: 
the virofunjfiarur/jLoi of Theodorus Metochita (d. 1332) who speaks c 
" Euclid of Megara, the Socratic philosopher, contemporary of Plato, 
as the author of treatises on plane and solid geometry, data, optic 
etc. : and a Paris MS. of the 14th century has " Euclidis philosoph 
Socratici liber elementorum.'' The misunderstanding was genera 
in the period from Campanus' translation (Venice 1482) to those c 
Tartaglia (Venice 1565) and Candalla (Paris 1566). But on 
Constantinus Lascaris (d. about 1493) had already made the prope 

^ Pappus, VII. pp. 676, 45 — 678, 6. Hultsch, it is true, brackets the whole possag 
pp. 676, 95—^78, 15, but apparently on the ground of the diction only. 
* Stobaeus, /,e. * viii. la, ext. i. 



^m 



4 INTRODUCTION [cH. i 

distinction by saying of our Euclid that " he was different from him 
of Megara of whom Laertius wrote, and who wrote dialogues*'' ; and 
to Commandinus belongs the credit of being the first translator' to 
put the matter beyond doubt : *' Let us then free a number of people 
from the error by which they have been induced to believe that our 
Euclid is the same as the philosopher of Megara" etc 

Another idea, that Euclid was born at Gela in Sicily, is due to the 
same confusion, being based on Diogenes Laertius' description' of the 
philosopher Euclid as being *'of Megara, or, according to some, of 
Gela, as Alexander says in the ^ea^o^^atV 

In view of the poverty of Greek tradition on the subject even as 
early as the time of Proclus (410-485 a.d,), we must necessarily take 
a*m grano the apparently circumstantial accounts of Euclid given by 
Arabian authors ; and indeed the origin of their stories can b^ 
explained as the result (i) of the Arabian tendency to romancep and 
(2) of misunderstandings. 

We read* that " Euclid, son of Naucrates, grandson of Zenarchus* 
called the author of geometry, a philosopher of somewhat ancient 
date, a Greek by nationality domiciled at Damascus, bom at Tyre, 
most learned in the science of geometry, published a most excellent 
and most useful work entitled the foundation or elements of geometry, 
a subject in which no more general treatise existed before among the 
Greeks :fnay, there was no one even of later date who did not walk 
in his footsteps and frankly profess his doctrine. Hence ako Greek, 
Roman and Arabian geometers not a few, who undertook the task 
of illustrating this work, published commentaries, scholia, and notes 
upon it, and made an abridgment of the work itself For this reason 
the Greek philosophers used to post up on the doors of their schools 
the weli*known notice : * Let no one come to our school, who has not 
first learned the elements of Euclid.*"! The details at the beginning 
of this extract cannot be derived from Greek sources, for even Proclus 
did not know anything about Euclid's father, while it was not the 
Greek habit to record the names of grandfathers, as the Arabians 
commonly did. Damascus and Tyre were no doubt brought in to 
gratify a desire which the Arabians always showed to connect famous 
Greeks in some way or other with the East Thus Naslraddlnj the 
translator of the Elemenis^ who was of Tus in Khurasan, actually 
makes Euclid out to have been "Thusinus" also* The readiness of 
the Arabians to run away with an idea is illustrated by the last words 

^ Letter to Fe maud us Acuna^ printed in Mauroljcus, HUiima SuiHa^f foL 91 r> (see 
Heiberg, Evklid-Studien^ pp. ^i — 3, 45), 
' Prcf»ce to translation (risauri, 'S?^)* 

* Diog. L. II. io6t p* 58 ed. Cobet. 

< Casiri, BiMiptkeca Arabka^Hispema SicuriaUmiiy I. p. 339* CasiH's source i» *1- 
Qlftl (d- 1343), the author of the Tt^rikh al-HukamSy a collection of biographies of phi- 
losophers, mathematicians, a^tronotncrs eta 

* The Fihrist says *'son of Naucratei, the son of Berenice (?) " {set Suter's translatLOO in 
Ahkandlungm ittr Gts<:h. d. Math. VI. Heft, 1893, p, 16 J, 

' The same predilection made the Ambs de^ritie Pythagoras as a P^P^^ ^^ ^^^ ^^ 
Salome, Hipmrchus as the exponent of Chaldaean philosophy or as ibe Chaldaean, Archi- 
medes as an Egyptian etc. (H&ji Khalfaj Lexicon Sibiic^apkuum, and Casiri). 






CH. i] EUCLID AND TRADITIONS ABOUT HIM 5 

of the extract Everyone knows the story of Plato's inscription over 
the porch of the Academy : " let no one unversed in geometry enter 
my doors " ; the Arab turned geometry into Euclid s geometry, and 
told the story of Greek philosophers in general and 'Hheir Academies." 

Equally remarkable are the Arabian accounts of the relation of 
Euclid and Apollonius\ According to them the Elements were 
originally written, not by Euclid, but by a man whose name was 
Apollonius, a carpenter, who wrote the work in 15 books or sections*. 
In the course of time some of the work was lost and the rest became 
disarranged, so that one of the kings at Alexandria who desired to 
study geometry and to master this treatise in particular first questioned 
about it certain learned men who visited him and then sent for Euclid 
who was at that time famous as a geometer, and asked him to revise 
and complete the work and reduce it to order. Euclid then re-wrote 
it in 13 books which were thereafter known by his name. (According 
to another version Euclid composed the 13 books out of commentaries 
which he had published on two books of Apollonius on conies and 
out of introductory matter added to the doctrine of the five regular 
solids.) To the thirteen books were added two more books, the work 
of others (though some attribute these also to Euclid) which contain 
several things not mentioned by Apollonius. According to another 
version Hypsicles, a pupil of Euclid at Alexandria, offered to the 
king and published Books xiv. and xv., it being also stated that 
Hypsicles had "discovered" the books, by which it appears to be 
suggested that Hypsicles had edited them from materials left by Euclid. 

We observe here the correct statement that Books XI v. and XV. 
were not written by Euclid, but along with it the incorrect informa- 
tion that Hypsicles, the author of Book xiv., wrote Book xv. also. 

The whole of the fable about Apollonius having preceded Euclid 
and having written the Elements appears to have been evolved out of 
the preface to Book xiv. by Hypsicles, and in this way ; the Book 
must in early times have been attributed to Euclid, and the inference 
based upon this assumption was left uncorrected afterwards when it 
was recognised that Hypsicles was the author. The preface is worth 
quoting : 

" Basilides of Tyre, O Protarchus, when he came to Alexandria 
and met my father, spent the greater part of his sojourn with him on 
account of their common interest in mathematics. And once, when 

^ The authorities for these statements (quoted by Casiri and Hajl Khalfa are al-KindI*s 
tract di imtituto Uhri Euclidis (al-Kindl died about 87^) and a commentary by (^ftd1z3de 
ar-RumI (d. about '440) on a book called Ashkdl at-ta: sis (fundamental propositions) by 
Ashraf Shamsaddin as-Samarqandl (^. 1176) consisting of elucidations of 35 propositions 
selected from the first books of Euclid. Na^Iraddln likewise says that Euclid cut out two of 
15 books of elements then existing and published the rest under his own name. According to 
Q&dlz&de the king heard that there was a celebrated geometer named Euclid at Tyre: Naflr- 
adciln says that he sent for Euclid of Tus. 

* So says the Fihrist, Suter (op! cit, p. 49) thinks that the author of the Fihrist did not 
suppose Apollonius of Perga to be the writer of the EUtnents^ as later Arabian authorities 
did, but that he distinguished another Apollonius whom he odls **a carpenter." Sutor's 
argument is based on the fact that the Fihrisfs article on Apollonius (of Perga) says nothins; 
of the Elements^ and that it gives the three great mathematicians, Euclid, Archimedes and 
Apollonius, in the correct chronological order. 

K 



A INTRODUCTION [cH. i 

examining the treatise written by Apollonius about the comparison 
between the dodecahedron and the icosahedron inscribed in the same 
sphere, (showing) what ratio they have to one another, they thought 
that Apollonius had not expounded this matter properly, and 
accordingly they emended the exposition, as I was able to learn 
from my father. And I myself, later, fell in with another book 
published by Apollonius, containing a demonstration relating to the 
subject, and I was greatly interested in the investigation of the 
problem. The book published by Apollonius is accessible to all— 
for it has a large circulation, having apparently been carefully written 
out later — ^but I decided to send you the comments which seem to 
me to be necessary, for you will through your proficiency in mathe- 
matics in general and in geometry in particular form an expert 
judgment on what I am about to say, and you will lend a kindly ear 
to my disquisition for the sake of your friendship to my father and 
your goodwill to me." 

The idea that Apollonius preceded Euclid must evidently have 
been derived from the passage just quoted. It explains other things 
besides, Basilides must have been confused with ffatrtX^vs, and we 
have a probable explanation of the "Alexandrian king,'* and of the 
"learned men who visited'* Alexandria. It is possible also that in 
the '* Tyrian '' of Hypsicles' preface we have the origin of the notion 
that Euclid was bom in Tyre, These inferences argue, no doubt, 
very defective knowledge of Greek : but we could expect no better 
from those who took the Organon of Aristotle to be " instrumentum 
musicum pneumaticum/' and who explained the name of Euclid, 
which they variously pronounced as Uclid^s or Icludes, to be com- 
pounded of Ucli a key, and Dis a measure, or, as some say, geometry, 
so that Udides is equivalent to the k^ of geametfyl 

Lastly the alternative version, given in brackets above, which says 
that Euclid made the Eiimenis out of commentaries which he wrote 
on two books of Apollonius on conies and prolegomena added to the 
doctrine of the five solids, seems to have arisen, through a like 
confusion, out of a later passage' in Hypsicles* Book xiv, : "And this 
is expounded by Aristaeus in the book entitled 'Comparison of the five 
figures,' and by Apollonius in the second edition of his comparison of 
the dodecahedron with the icosahedron.'* The "doctrine of the five 
solids " in the Arabic must be the " Comparison of the five figures " 
in the passage of Hypsicles, for nowhere else have we any information 
about a work bearing this title, nor can the Arabians have had. The 
reference to the tivo books of Apollonius on conks will then be the 
result of mixing up the fact that Apollonius wrote a book on conies 
with the second edition of the other work mentioned by Hypsicles. 
We do not find elsewhere in Arabian authors any mention of a 
commentary by Euclid on Apollonius and Aristaeus : so that the 
story in the passage quoted is really no more than a variation of the 
fable that the Elements were the work of Apollonius, 

^ Heiberg's Eaclld, toL v* p. 6, 



f 



CHAPTER II. 



y EUCLID'S OTHER WORKS. 

In giving a list of the Euclidean treatises other than the EUmeHts, 
I shall be brief: for fuller accounts of them, or speculations with 
regard to them, reference should be made to the standard histories of 
mathematics ^ 

I will take first the works which are mentioned by Greek authors. 

I. The Pseudaria, 

I mention this first because Proclus refers to it in the general 
remarks in praise of the Elements which he gives immediately after 
the mention of Euclid in his summary. He says': "But, inasmuch 
as many things, while appearing to rest on truth and to follow from 
scientific principles, really tend to lead one astray from the principles 
and deceive the more superficial minds, he has handed down methods 
for the discriminative understanding of these things as well, by the 
use of which methods we shall be able to give beginners in this study 
practice in the discovery of paralogisms, and to avoid being misled. 
This treatise, by which he puts tiiis machinery in Our hands, he 
entitled (the book) of Pseudaria, enumerating in order their various 
kinds, exercising our intelligence in each case by theorems of all 
sorts, setting the true side by side with the false, and combining 
the refutation of error with practical illustration. This book then is 
by way of cathartic and exercise, while the Elements contain the 
irrefragable and complete guide to the actual scientific investigation 
of the subjects of geometry." 

The book is considered to be irreparably lost We may conclude 
however from the connexion of it with the Elenunts and the reference 
to its usefulness for beginners that it did not go outside the domain 
of elementary geometry*. 



^ Heibeig gives very exhaustive details in his LitterargeschichtlUhe Studien iiber Euklid\ 
the best of the shorter accounts are those of Cantor (Gesch, d. Afaih. i,, 1907, pp. 378 — 394) 
and Loria (II periodo aureo delta geometria greca^ p. 9 and pp. 63 — 85). 

• Proclus, p. 70, I— 18. 

' Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his 
commentary on Aristotle's Sophistici EUnchi (fol. 15 bV. **Not only those (AeTxoc) which do 
not start from the principles of the science, under which the problem is classed... but also 
those which do start from the proper principles of the science but in some respect admit a 
paralogism, e.g. the Fseudographanata of Euclid." Tannery (BuU. des sciences math, et astr, 
1* S^e, vi., 1883, i^ Partie, p. 147) conjectures that it may be from this treatise that the 
same commentator got his information about the quadratures of the circle by Antiphon and 



INTRODUCTION [cH. ii 



r 



2. The Data 

The Data {Z€topi,iifa) are included by Pappus in the Treasury af 
Analysis {tqitok ava\v6^^o^\ and he describes their contents*. They 
are still concerned with elementary geometry, though forming part 
of the introduction to higher analysis. Their form is that of pro- 
positions proving that, if certain things in a figure are given (in 
magnitude, in species, etc.)* something else is given. The subject* 
matter Is much the same as that of the plani metrical books of the \ 
Elements, to which the Data are often supplementary. We shall see 
this later when we come to compare the propositions in the Elements 
which give us the means of solving the general quadratic equation 
with the corresponding propositions of the Data which give the 
solution. The Data may in fact be regarded as elementary exercises 
in analysis. 

It is not necessary to go more closely into the contents, as we 
have the full Greek text and the commentary by Marinus newly 
edited by Menge and therefore easily accessible* 

3- The book On divisions {ofjignres). 

This work {ir^pl Biaif>icr€wv fit^iov) is mentioned by Proclus*. 
In one place he is speaking of the conception or definition {X0709) 
o{ figure, and of the divisibility of a figure into others differing from 
it in kind ; and he adds: "For the circle is divisible into parts unlike 
in definition or notion {dvofiGta t^ >Jrf^\ and so is each of the 
rectilineal figures ; this is in fact the business of the writer of the 
Elements in his Divisions, where he divides given figures, in one case 
into like figures, and in another into unlike**" "Like" and "unlike" 
here mean, not "similar*' and "dissimilar** in the technical sense, but 
"like" or "unlike in definition or notion'' (Xo^^): thus to divide a 
triangle into triangles would be to divide it into "like" figures, to 
divide a triangle into a triangle and a quadrilateral would be to 
divide it into "unlike" figures. 

The treatise is lost in Greek but has been discovered in the 
Arabic. First John Dee discovered a treatise De divisionibus by one 
Muhammad Bagdad] n us ^ and handed over a copy of it (in Latin) in 
1563 to Commandinus, who published it, in Dee's name and his own, 
in IS70*, It was formerly supposed that Dee had himself translated 

BrysoD, to say aothing of the lunules of Hippocrates* I tliink however that there b an 
objection to this theory so far as regank Brysooi for Alexander distinctly lays that Biyson^s 
quadrature did not start from the proper priijcipJes of geometfy, hut frofn some priiuriples 
more generat. 

" Pappus, vn. p. 638. 

' VoL VI ♦ in the Teiibner edition of EuctMii cftra otnnm by Heiberg and Menge. A 
translation of the Puta is also included In Simson a EucHd (though culturally his text left 
much to be desired)* 

■ Proclusj p. 69 J 4. * ihid. J 44, 21— 16- 

* Steinschneider places him in the totlic. H. Sv^itt {BthH^ktca Maihtmaika, iVj, 1903, 
pp. 34, 17) ideoti^es him with Abu(Bekr) Moh. b. "Abdalbiql al-Ba^adl, QSdl 0ud^ey of 
M&rbtiln {drca lojo-n+t), to whom he also attributes the Libtrj^dti (? judicis) super dtamu^ 
Euelidii translated by Gherard of Cremona. 

* Df superfHtrum ditnsionihus liher Math&mdo Bagitadina aditriptui^ nurv primttm 
Uannis Dtt Londinensii it Fed^^i C^mfrmttdini Urhinntis eptra itt iu€fm tiMhiSi Plsaurt, 
1570, afterwania bduded in Gregory's Euclid {Oxford, 1703)- 



CH. ii] EUCLID'S OTHER WORKS 9 

the tract into Latin from the Arabic*; but it now appears certain' 
that he found it in Latin in a Cotton MS. now in the British Museum. 
Dee, in his preface addressed to Commandinus, says nothing of his 
having translated the book, but only remarks that the very illegible 
MS. had caused him much trouble and (in a later passage) speaks of 
••the actual, very ancient, copy from which I wrote out..** (in ipso 
unde descripsi vetustissimo exemplari). The Latin translation of 
this tract from the Arabic was probably made by Gherard of Cremona 
(1114-1187), among the list of whose numerous translations a "liber 
divisionum" occurs. The Arabic original cannot have been a direct 
translation from Euclid, and probably was not even a direct adapta- 
tion of it ; it contains mistakes and unmathematical expressions, and 
moreover does not contain the propositions about the division of a 
circle alluded to by Proclus. Hence it can scarcely have contained 
more than a fragment of Euclid's work. 
\ But Woepcke found in a MS. af Paris a treatise in Arabic on the 

division of figures, which he translated and published in 1851*. It is 
expressly attributed to Euclid in the MS. and corresponds to the 
description of it by Proclus. Generally speaking, the divisions are 
divisions into figures of the same kind as the original figures, e.g. of 
triangles into triangles; but there are also divisions into " unlike*' 
figures, e.g. that of a triangle by a straight line parallel to the base. 
The missing propositions about the division of a circle are also here : 
**to divide into two equal parts a given figure bounded by an arc 
of a circle and two straight lines including a given angle" and "to 
draw in a given circle two parallel straight lines cutting off a certain 
part of the circle." Unfortunately the proofs are given of only four 
propositions (including the two last mentioned) out of 36, because 
the Arabic translator found them too easy and omitted them. To 
illustrate the character of the problems dealt with I need only take 
one more example : " To cut off a certain fraction from a (parallel-) 
trapezium by a straight line which passes through a given point lying 
inside pr outside the trapezium but so that a straight line can be 
drawn through it cutting both the parallel sides of the trapezium." 
The genuineness of the treatise edited by Woepcke is attested by the 
facts that the four proofs which remain are elegant and depend on 
propositions in the Elements, and that there is a lemma with a true 
Greek ring: "to apply to a straight line a rectangle equal to the 
rectangle contained hy AB, AC and deficient by a square'' Moreover 
the treatise is no fragment, but finishes with the words " end of the 
treatise," and is a well-ordered and compact whole. Hence we may 
safely conclude that Woepcke's is .not only Euclid's own work but 
the whole of it*. A restoration of the work, with proofs, was attempted 

^ Heiberg, Euklid'Studien, p. 13. 

' H. Suter in Bibliotkeca MaihematUa, iv,, 1905-6, pp. 331—3. 

• Journal Asiatique, 185 1, p. 133 sgq. 

* We are told by Casiri that Th&bit b. Qarra emended the translation of the Uber de 
divisumibus\ but Ofterdinger seems to be wrong in sajring that according to Gartz (/?/ i$Uer- 
prUUms et expianaioribus EuciuUs AraHcis schediasma histancum, HaUe, 1813) there is a 



10 INTRODUCTION [ch, h 

by OfterdingerS who however does not give Woepckc's props. 30, 31, 

34* 35i 3^^ 

4. The Porisms. 

It is not possible to give in this place any account of the con- 
troversies about the contents and signiEcance of the three lost books 
of Porisms, or of the important attempts by Robert Simson and 
Chasles to restore the work. These may be said to form a whole 
literature, references to which will be found most abundantly given 
by Heiberg and Loria, the former of whom has treated the subject 
from the philological point of view; most exhaustively, while the 
latter, founding himself generally on Heiberg, has added useful 
detailsj from the mathematical side, relating to the attempted restora- 
tions, etc,' It must suffice here to give an extract from the only 
original source of information about the nature and contents of the 
P^nsms, namely Pappus', In his general preface about the books 
composing the Treasury of Afmfysis (totto? aWXvd^evo^) he says : 

"After the Tangencies (of Apollonius) come, in three books, the 
Porisms of Euclid, [in the view of many] a collection most ingeniously 
devised for the analysis of the more weighty problems, [and] although 
nature presents an unlimited number of such porisms', [they have 
added nothing to what was written originally by Euclid, except that 
some before my time have shown their want of taste by adding to a 
few (of the propositions) second proofs^ each (proposition) admitting 
of a definite number of demonstrations, as we have shown, and 
Euclid having given one for each, namely that which is the most 
lucid* These porlsms embody a theory subtle, natural, necessary, 
and of considerable generality, which is fascinating to those >vho can 
see and produce results]. 

** Now all the varieties of porisms belong, neither to theorems nor 
problems, but to a species occupying a sort of intermediate position 
[so that their enunciations can be formed like those of either theorems 
or problems], the result being that, of the great number of geometers, 
some regarded them as of the class of theorems, and others of pro- 
blems, looking only to the form of the proposition. But that the 
ancients knew better the difference between these three things, is 
clear from the 'definitions* For they said that a theorem is that 
which is proposed with a view to the demonstration of the very 
thing proposed J a problem that which is thrown out with a view to 
the construction of the very thing proposed, and a porism that which 
is proposed with a view to the producing of the very thing proposed. 
[But this definition of the porism was changed by the more recent 
writers who could not produce everything, but used these elements 

CDnapkt« MS. of Tfaabit'9 tmnslAtioQ m ihe Escurml. I cannot find any such staCemcDt in 
Gaitz. 

^ L. F. Oftcvdingeri Stiirc^ tur WUdfrkersi£liung tUr Sckrifl dei Euklides iiber dk 
Tkdilung d^ Figurm^ Ulm, 185^- 

' Heiberg, Eukiid-S(udien^ pp, 56^ — 79, uwl Loha, H pftiodo tmrea ddla geometric gt^tif 

pp. 70— Si, 13J— 5. 

' pAppO£, cd. HuJudi, VII, pp, 648—660. I put in square brackeu the words bracket^ 
bj HuJtsck 

* I adopt Heibefg'a reading of a comma here instead of a full stop* — ^ * 






CH. n] EUCLID'S OTHER WORKS ii 

and proved only the fact that that which is sought really exists, but 
did not produce it^ and were accordingly confuted by the definition 
and the whole doctrine. They based their definition on an incidental 
characteristic, thus : A porism is that which falls short of a locus- 
theorem in respect of its hypothesis*. Of this kind of porisms loci 
are a species, and they abound in the Treasury of Analysis ; but 
this species has been collected, named and handed down separately 
from the porisms, because it is more widely diffused than the other 
species]. But it has further become characteristic of porisms that, 
owing to their complication, the enunciations are put in a contracted 
form, much being by usage left to be understood; so that many 
geometers understand them only in a partial way and are ignorant of 
the more essential features of their contents. 

^[Now to comprehend a number of propositions in one enunciation 
is by no means easy in these porisms, because Euclid himself has not 
in fact given many of each species, but chosen, for examples, one or a 
few out of a great multitude^ But at the beginning of the first book 
he has given some propositions, to the number of ten, of one species, 
namely that more fruitful species consisting of loci.] Consequently, 
finding that these admitted of being comprehended in one enunciation, 
we have set it out thus: 

If, in a system of four straight lines^ which cut each other 
two and two, three points on one straight line be given while the 
rest except one lie on different straight lines given in position, 
the remaining point also will lie on a straight line given in 
position'. 

^ Heiberg points out that Props. 5—^ of Archimedes' treatise On Sfirals are porisms in 
this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K, 
It is then pK)6sible, says Archimedes, to araw a straight line q g 

KHF^ meeting the circumference in H and the tangent in /*, 

such that 

FH\HK<(%szBiri\c, 

where c is the circomference of any circle. To prove this he 
assumes the following construction. E being any straight line 
greater than r, he says : let KG be parallel to DF^ "and let 
the line GH equal to i? be placed verging to the point B.^^ 
Archimedes must of course nave known how to effect this 
construction, which requires conies. But that it is possible requires very little argument, for 
if we draw any straight line BHG meeting the circle in ZTand KG in G^ it is obvious that 
as G moves away from C, HG becomes neater and greater and may be made as great as we 
please. The '* later writers *' would no doubt have contented themselves with this considera- 
tion without actually constructing HG, 

' As Heibere says, this translation is made certain by a preceding passage of Pappus 
0>> 648, I — 3) where he compares two enunciations, the latter of which *' falls short of the 
former in hypothesis but goes oeyond it in requirement,^'* E.g. the first enunciation requiring 
us, given three circles, to draw a circle touching all three, the second may require us, given 
only two circles (one less datum), to draw a circle touching them and of a given siie (an 
extra requirement). 

* I translate Heiberg's reading with a full stop here followed by rpbt dffxv ^ ^h*"^ l""/^ 
^}dt^ {ifiofjJpop) Hultsch] Tov vpibrou fiipxiov..., 

* The four straight lines are described in the text as (the sides) hrrlov 4 wapvwrlov, Le. 
sides of two sorts of quadrilaterals which Simson tries to explain (see p. no of the Index 
GraecitcUis of Hultsch*s edition of Pappus). 

* In other words (Cbasles, p. «3 ; Loria, p. 73) if a triangle be so deformed that each of 
its sides turns about one of three points in a straight line, and two of its vertices lie on two 
straight lines given in position, the third vertex will klso lie on a straight line. 




tJ INTRODUCTION [ch. ii 

**Thts has only been enunciated of four straight line*!, of which not 
more than two pass through the same point, but it is not known (to 
most people) that it is true of any assigned number of straight lines 
if enunciated thus: 

If any number of straight lines cut one another, not more 
than two (passing) through the same point, and all the points 
(of intersection situated) on one of them be given, and if each of 
those which are on another (of them) lie on a straight line given 
in position — 
or still more generally thus : 

if any number of straight lines cut one another, not more than 

two (passing) through the same point, and all the points (of 

intersection situated) on one of them be given, while of the other 

points of intersection in multitude equal to a triangular number 

a number corresponding to the side of this triangular number tie 

respectively on straight lines given in position, provided that of 

these latter points no three are at the angular points of a triangle 

{sc. having for sides three of the given straight lines)— each of the 

remaining points will lie on a straight h'ne given in position*. 

" It is probable that the writer of the Elements was not unaware 

of this but that he only set out the principle ; and he seems, in the 

case of all the porisms, to have laid down the principles and the 

seed only [of many important things], the kinds of which should be 

distinguished according to the differences, not of their hypotheses, but 

of the results and the things sought [All the hypotheses are different 

from one another because they are entirely special, but each of the 

results and things sought, being one and the same, follow from many 

different hypotheses.] 

"We must then in the first book distinguish the following kinds of 
things sought : 

"At the beginning of the book* is this proposition : 

I, ' If from two givm points straight lines be drawn meeting 
on a straight line given in position^ and one cut off from a straight 
Urn given in position (a segment measured) to a given point on i/, 
the other wilt also cut off from afwther {straight line a segment) 
having to the first a given ratio' 
*■ Following on this (we have to prove) 

IL that such and such a point lies on a straight line given 

in position ; 
II L that the ratio of such and such a pair of straight lines 
is given ; '* 
etc. etc. (up to xxix.). 
^The three books of the porisms contain 38 lemmas; of the 
theorems themselves there are 171." 

^ Loria (p. 7^ note) gives the meaning of this as follows, pointing out that Simson was 
the discoverer o( it : " Iia complete M-lateral be deformed so that its sides respectively turn 
about n points on a straight line, and (it- i) of its n (n- i)/a vertices move on as many 
straight Imes, the other (m- i)(ii-9)/a of its vertices likewise move on as many straight 
lines: but it is necessary that it shoula be impossible to form with the (m- i) vertices any 
triangle having for sides the sides of the polygon." 

' Reading, with Keiberg, roH fitfiXlov [roC r Hultsch]. 



\ 



CH. ii] EUCUiyS OTHER WORKS 13 

Pappus further gives lemmas to the Porisms (pp. 866 — 918, ed, 
Hultsch). 

With Pappus' account of Porisms must be compared the passages 
of Proclus on the same subject Proclus distinguishes two senses in 
which the word ir6pur/jui is used. The first is that of corollary where 
something appears as an incidental result of a proposition, obtained 
without trouble or special seeking, a sort of bonus which the investi- 
gation has presented us with^ The other sense is that of Euclid's 
Porisf9ts\ In this sense' ^^porism is the name given to things which 
are sought, but need some finding and are neither pure bringing into 
existence nor simple theoretic argument For (to prove) that the 
angles at the base of isosceles triangles are equal is a matter of 
theoretic argument, and it is with reference to things existing that 
such knowl^ge is (obtained). But to bisect an angle, to construct a 
triangle, to cut off, or to place — all these things demand the making 
of something ; and to find the centre of a given circle, or to find the 
greatest common measure of two given commensurable magnitudes, 
or the like, is in some sort between theorems and problems. For in 
these cases there is no bringing into existence of the things sought, 
but finding of them, nor is the procedure purely theoretic. For it is 
necessary to bring that which is sought into view and exhibit it to 
the eye. Such are the porisms which Euclid wrote, and arranged in 
three books of Porisms. 

Proclus' definition thus agrees well enough with the first, ** older," 
definition of Pappus. A porism occupies a place between a theorem 
and a problem: it deals with something dXreaidy existing, ^s a theorem 
does, but has to find it (e.g. the centre of a circle), and, as a certain 
operation is therefore necessary, it partakes to that extent of the 
nature of a problem, which requires us to construct or produce some- 
thing not previously existing. Thus, besides IIL i of the Elements 
and X. 3, 4 mentioned by Proclus, the following propositions are 
real porisms: ill. 25, VL 11— 13, vii. 33, 34, 36, 39, vill. 2,4, x. 10, 
XIIL 18. Similarly in Archimedes On the Sphere and Cylifuler I. 2 — 6 
might be called porisms. 

The enunciation given by Pappus as comprehending ten of Euclid's 
propositions may not reproduce the form of Euclid's enunciations ; 
but, comparing the result to be proved, that certain points lie on 
straight lines given in position, with the class indicated by ll. above, 
where the question is of such and such a point lying on a straight line 
given in position, and with other classes, e.g. (v.) that such and such a 
line is given in position, (vi.) that such and such a line verges to a given 
point, (XXVII.) that there exists a given point such that straight lines 
drawn from it to such and such (circles) will contain a triangle given 
in species, we may conclude that a usual form of a porism was ^* to 
prove that it is possible to find a point with such and such a property" 

* Proclus, pp. a I a, 14 ; 301, «i. 

' ibid. p. 313, I a. "The term porism is used of certain problems, like the Porisms 
written by Euclid." 

* ihid, pp. 301, 35 sqq. 



14 INTRODUCTION [CB. ii 

or "a straight line on which lie all the points satisfying given 
conditions" etc. 

Simson defined a porism thus : " Porisma est propositio iti qua 
proponitur dem oust rare rem aliquam, vel pi u res datas esse^ cuij vel 
quibus, ut et cuilibet ex rebus innumeriSj non quidem datis, sed quae 
ad ea quae data sunt eandem habent relationem, con venire ostendendum 
est affecttonem quandam communem in propositione descriptamV 

From the above it is easy to understand Pappus' statement that 
loci constitute a large class of porisms. A hcus is well defined by 
Simson thus : '* Locus est propositio in qua propositum est datam 
esse demonstrare, vel in venire lineam aut superficiem cuius quodlibet 
punctum, vel superficiem in qua quaelibet linea data lege descripta, 
communem quandam habet proprietatem in propositione descriptam." 
Heiberg cites an excellent instance of a l<fats which is Sip&rism, namely 
the following proposition quoted by Eutocius* from the P/afie Zati of 
ApoHonius : 

*' Given two points in a plane, and a ratio between unequal straight 
lines, it is possible to draw, in the plane, a circle such that the straight 
lines drawn from the given points to meet on the circumference of 
the circle have (to one another) a ratio the same as the given ratio/* 

A difficult point, however, arises on the passage of Pappus, which 
says that a porism is "that which, in respect of its hypothesis, falls 
short of a locus-theorem" (roTrueov ^€6>/j?J^aTo?)- Heiberg explains it 
by comparing the porism from ApoHonius' Platit Xa^i just given with 
Pappus' enunciation of the same thing, to the effect that, if from two 
given points two straight lines be drawn meeting in a point, and these 
straight lines have to one another a given ratio, the point will lie on 
either a straight line or a circumference of a circle given in position. 
Heiberg observes that in this latter enunciation something is taken 
into the hypothesis which was not in the hypothesis of the enunciation 
of the porism, viz. "that the ratio of the straight lines is the same." 
I confess this does not seem to me satisfactory : for there is no real 
difference between the enunciations, and the supposed difference in 
hypothesis is very like playing with words. Chasles says : " Ct qui 
const itu€ U porisme est ct qui manque d I'hypoth^e d'un tASreme 
local (en d 'aut res termes, le porisme est infifrieur, par Vhypoth^se, au 
th^orimc local; c'est-i*dire que quand quelques parties d'une pro- 
position locale n'ont pas dans T^nonc^ la determination qui leur est 
propre, cette proposition cesse d'etre regardee com me un th^rfeme et 
devient un porisme)/* But the subject still seems to require further 
elucidation. 

While there is so much that is obscure, it seems certain (i) that the 
Porisms were distinctly pait of higher geometry and not of elementary 

^ This was thus expressed bj Cbasles i '* Le porisme ^ udc propositioti dons laquellc on 
dcmimde de d^montrer qu'unc chose ou plusieurs choses sont d^nn/es, qui, flinsi que iWe 
quelcouque d'une in^nUe d^autres choses non donates, mais dont chacune est avec des choses 
doDufes daoa u&e m^me relation, one une cert^ine propri^t^ comoiune, d^rite dam Ja pio^ 
position, " 

■ Commentary on ApoUonius* C^ni^s (vol. ii. p, iSo^ ed. Heiberg). 



1 



I* 



CH. II] EUCLID'S OTHER WORKS 15 

geometry, (2) that they contained propositions belonging to the 
modem theory of transversals and to projective geometry. It should 
i be remembered too that it was in the course of his researches on this 
subject that Chasles was led to the idea of anharmanic ratios. 

Lastly, allusion should be made to the theory of Zeuthen^ on the 
subject of the porisms. He observes that the only porism of which 
Pappus gives the complete enunciation, *' If from two given points 
straight lines be drawn meeting on a straight line given in position, 
and one cut off from a straight line given in position (a segment 
measured) towards a given point on it, the other will also cut off from 
another (straight line a segment) bearing to the first a given ratio," 
is also true if there be substituted for the first given straight line a 
conic regarded as the '' locus with respect to four lines," and that this 
extended porism can be used for completing Apollonius' exposition 
of that locus. Zeuthen concludes that the Porisms were in part by- 
products of the theory of conies and in part auxiliary means for the 
study of conies, and that Euclid called them by the same name as 
that applied to corollaries because they were corollaries with respect to 
conies. But there appears to be no evidence to confirm this conjecture. 

5. The Surface-loci (tmto* irpo^ em<f>aif€ia). 

The two books on this subject are mentioned by Pappus as part 
of the Treasury of Analysis^. As the other works in the list which 
were on plane subjects dealt only with straight lines, circles, and 
conic sections, it is a priori likely that among the loci in this treatise 
(loci which are surfaces) were included such loci as were cones, 
cylinders and spheres. Beyond this all is conjecture based on two 
lemmas given by Pappus in connexion with the treatise. 

(i) The first of these lemmas' and the figure attached to it are 
not satisfactory as they stand, but a possible restoration is indicated 
by Tannery*. If the latter is right, it suggests that one of the loci 
contained all the points on the elliptical parallel sections of a cylinder 
and was therefore an oblique circular cylinder. Other assumptions 
with r^ard to the conditions to which the lines in the. figure may be 
subject would suggest that other loci dealt with were cones regarded 
as containing all points on particular elliptical parallel sections of 
the cones*. 

(2) In the second lemma Pappus states and gives a complete proof 
of the focus-and-directrix property of a conic, viz. that the locus of a 
point whose distance from a given point is in a given ratio to its distance 
from a fixed line is a conic section^ which is an ellipse ^ a parabola or a 
hyperbola according as the given ratio is less than, equal to, or greater 
than unity\ Two conjectures are possible as to the application of 
this theorem in Euclid's Surface-loci, (a) It may have been used to 
prove that the locus of a point whose distance from a given straight 

^ Die Lehre von den Ktgdschnitten im AUertum^ chapter viii. 

• Pappus, VII. p. 636. • ibid. vii. p. 1004. 

* Bulletin des sciences math, ei astron.., 3* S^rie, vi. 149. 

• Further particulars will be found in The IVarks e/ Archimedes, pp. Ixii — Ixiv, and in 
Zeuthen, Die Lehre von den JCegeischnitten^ p. 415 sqq. 

* Pappus, VII. pp. 1006 — 1014, and Hultsch*s Applendix, pp. 1370 — 3. 



i6 INTRODUCTION - [CH. ii 

line is in a given ratio to its distance from a given plane is a certain 
cone. {6} It may have been used to prove that the locus of a point 
whose distance from a given point is in a given ratio to its distance 
from a given plane is the surface formed by the revolution of a conic 
about its major or conjugate axis^ Thus Chasles may have been 
correct in his conjecture that the Surfaa-ioci dealt with surfaces of 
revolution of the second degree and sections of the same^ 

6, The Conks. 

Pappus says of this lost work; **The four books of Euclid's Conies 
were completed by ApoUonius, who added four more and gave us 
eight books of Conies'.*' It is probable that Euclid's work was lost 
even by Pappus' time, for he goes on to speak of '* Aristaeus, who wrote 
the still extant five books of Solid Loci connected with the conies/' 
Speaking of the relation of Euclid's work to that of Aristaeus on conies 
regarded as loci ^ Pappus says in a later passage (bracketed however 
by Hultsch) that Euclid, regarding Aristaeus as deserving credit for 
the discoveries he had already made in conies, did not (try to) 
anticipate him or construct anew the same system. We may no 
doubt conclude that the book by Aristaeus on solid loci preceded 
Euclid's on conies and was, at least in point of originality, more 
important Though both treatises dealt with the same subject-matter^ 
the object and the point of view were different ; had they been the 
same, Euclid could scarcely have refrained, as Pappus says he did, 
from attempting to improve upon the earlier treatise* No doubt 
Euclid wrote on the general theory of conies as Apollonius did, but 
confined himself to those properties which were necessary for the 
analysis of the Solid Loci of Aristaeus. The Conies of Euclid were 
evidently superseded by the treatise of Apollonius, 

As regards the contents of Euclid's Conies^ the most important 
source of our information is Archimedes, who frequently refers to 
propositions in conies as well known and not needing proof, adding 
in three cases that they are proved in the " elements of conies '* or in 
'*the conies," which expressions must clearly refer to the works of 
Aristaeus and Euclid** 

Euclid still used the old names for the conies (sections of a right- 
angled» acute*angled, or obtuse -angled cone), but he was aware that 
an ellipse could be obtained by cutting a cone in any manner by a 
plane parallel to the base (assuming the section to lie wholly between 
the apex of the cone and its base) and also by cutting a cylinder. 
This is expressly stated in a passage from the Phaenomma of Euclid 
about to te mentioned*, 

7* The Phaenonuna. 

This is an astronomical work and is still extant A much inter- 

* For further d^taib sec Tlis Worki of Attkimidis, pp, bdv, latv, aod Zeuthen, /♦ f» 

* A^fH Misi^riyuff pp, 373—4' * Pappus, vri* p. 671. 

* For details of these proposition* see my Apoiknims 0/ Ptrga^ pp. stxiv, xnjtvi. 

' See Heitjerg, Mukiid*StudUn^ p. 88. " If a cone or a cylinder be cut by a pkiae not 
parallel to the base» the section b a section of an acute -wigled conc^ which is like a shield 



'! CH.n] EUCLID'S OTHER WORKS 17 

'( polated version appears in Gregor/s Euclid, and a much earlier and 

• f better recension is, says Heiberg\ contained in the MS. Vindobonensis 
M philos. Gr. 103, though the end of the treatise, from the middle of 

I prop. 16 to the last (18), is missing. The book consists of i8 pro- 
positions of j;^A^rrir geometry. Euclid based it on Autolycus' work 

y wepl tcipovfiipff^ a^loa^y but also, evidently, on an earlier textbook of 
Sphaerka of exclusively mathematical content. It has been con- 

I jectured that the latter textbook may have been due to Eudoxus^ 

* 8. The Optics. 

This book needs no description, as it has been edited by Heiberg 
recently*, both in its genuine form and in the recension by Theon. 
The Cataptrica published by Heiberg in the same volume is not 
genuine, and Heiberg suspects that in its present form it may be 
Theon 's. Il^is not even certain that Euclid wrote Cataptrica at 
all, as Proclus may easily have had Theon's work before him and 
inadvertently assigned it to Euclid^. 

9. Besides the above-mentioned works, Euclid is said to have 
written the Elements of Music* (ai xara fiovaiK^v oTotveiworcA?). Two 
treatises are attributed to Euclid in our MSS. of the Musici, the 
KaraTOfifj mapovo^, Sectio canonis (the theory of the intervals)*, and the 
elaaycaiyif apfLOviKi^ (introduction to harmony). The first, resting on 
the Pythagorean theory of music, is mathematical and clearly and well 
written, the style and the form of the propositions agreeing well with 
what we find in the Elements, Its genuineness is confirmed not only 
by internal evidence but by the fact that almost the whole of the 
treatise (except the preface) is quoted in extenso, and Euclid is twice 
mentioned by name, in the commentary on Ptolemy's Harmonica 
published by Wallis and attributed by him to Porphyry, but probably 
for the most part compiled by Pappus or some other competent 
mathematician'. (On the other hand Tannery set himself to prove 
that the treatise is not authentic'.) The second treatise is not Euclid's, 
but was written by Cleonides, a pupil of Aristoxenus*. 

Lastly, it is worth while to give the Arabians* list of Euclid's 
works. I take this from Suter's translation of the list of philosophers 
and mathematicians in the Fihrist^ the oldest authority of the kind 
that we possess**. "To the writings of Euclid belong further [in 
addition to the Elements^ : the book of Phaenomena ; the book of 



* Euklui'StudUn^ pp. 50 — I. 
' Heiberg, op. cit, p. 46 ; Hultsch, Autolycus^ p. xii ; A. A. Bjornbo, Studien iiber 

i Menelaos' Stkdrik (Abhandlungen %ur GtschUhte der mathemcUischen WissenschafUn^ XI v. 

1901), p. 56 sqq. 
1 • Etulidis opera omnia^ vol. vii. (1895). 

* Heiberg, Euclid's Optics^ etc, p. 1. • Proclus, p. 69, 3. 

* Published in the Musici Scriptores Graeci^ ed. Jan (Teubner, 1895), pp. 113— 166. 
' Jan, Musici Scriptores Gtaeci^ p. 116. 

* Comptes rendus de VAcad, des inscriptions et belles-lettres y Paris, 1904, pp. 439^445* 
Cf. Bibliotheca Mathemaiicay vi,, 1905-6, p. 325, note i. 

* Heiberg, Euklid'Studien, pp. 52—5; Jan, Musici Scriptores Graeci, pp. 169 — 174. 
*• H. Suter, Dcu McUhematiker' Verzeichniss im Fihrist in Abhandlungen %ur Geschichte 

dir McUkematiky VI., 189a, pp. i — 87 (see especially p. 17). Cf. Casiri, I. 339, 340, and 
Garti, pp. 4, 5. 

H. E. 2 



i8 



INTRODUCTION 



[CH, II 



Given Magnitudes [Data]; the book of Tones, known under the name 
of Music, not genuine; the book of Division, emended by Thabit; 
the book of Utilisations or Applications [Parisms\ not genuine; the 
book of the Canon ; the book of the Heavy and Light ; the book of 
Synthesis, not genuine; and the book of Analysis, not genuine/' 

It is to be observed that the Arabs already regarded the book of 
Tones (by which must be meant the tla^ayoryfi appi^oviKiq) as spurious. 
The book of Division is evidently the book on Divisions {of figures). 
The next book is described by Casin as " liber de utilitate suppositus." 
Sutcr gives reason for believing the Poristns to be meant', but does 
not apparently offer any explanation of why the work is supposed to 
be spurious. The book of the Canon is clearly the Kara-rop.ri mavov^^i. 
The book on *' the Heavy aVid Light '' is apparently the tract t>e itvi 
a pondtrosOy included in the Basel Latin translation 0^1537, and in 
Gregory's edition. The fragment, however, cannot safely be attributed 
to Euclid, for (1) we have nowhere any mention of his having written 
on mechanics, (2) it contains the notion of speci6c gravity in a form 
so clear that it could hardly be attributed to anyone earlier than 
Archimedes'. Suter thinks* that the works on Analysis and Synthesis 
(said to be spurious in the extract) may be further developments of 
the Data or Parisms^ or may be the interpolated proofs of EucL 
Xin. I — ^s» divided into analysis uj\A synt^t^is, as to which sec the notes 
on those propositions. 

' Suter, <^« Hi, pp. 49, 50. Wenncli trAn&lai^ tb< word as "atUk." Suier i«^s that 

the n^rest ttiea^ning of tlie Arabic word u of "porism" is utt^ gain (NuUen, GcwionJ, while 
A further meaning is explan anions observation^ addition : a gain arising out of what has 
preceded (cT. Proclus' dentition of the porism in the sense of a corollary). 
* Heiberg, Euklid-Seudifn, pp. 9, jo» * Suter, ^, cU, p. 50. 



CHAPTER III. 

GREEK COMMENTATORS ON THE ELEMENTS OTHER 
THAN PROCLUS. 

That there was no lack of commentaries on the Elements before 
the time of Proclus is evident from the terms in which Proclus refers 
to them; and he leaves us in equally little doubt as to the value 
which, in his opinion, the generality of them possessed. Thus he says 
in one place (at the end of his second prologue)^ : 

"Before making a beginning with the investigation of details, 
I warn those who may read me not to expect from me the things 
which have been dinned into our ears ad nauseam (SiaredpvXfjTai) by 
those who have preceded me, viz. lemmas, cases, and so forth. For 
I am surfeited with these things and shall give little attention to them. 
But I shall direct my remarks principally to the points which require 
deeper study and contribute to the sum of philosophy, therein emulating 
the Pythagoreans who even had this common phrase for what I mean 
' a figure and a platform, but not a platform and sixpfrnr***'" — 

In another place* he says : " Let us now turn to the elucidation 
of the things proved by the writer of the Elements, selecting the more 
subtle of the comments made on them by the ancient writers, while 
I cutting down their interminable diflfuseness, giving the things which 
are more systematic and follow scientific methods, attaching more 
importance to the working-out of the real subject-matter than to the 
variety of cases and lemmas to which we see recent writers devoting 
'.themselves for the most part" 

At the end of his commentary on Eucl. I. Proclus remarks* that 
^the commentaries then in vogue were full of all sorts of confusion, and 
K:ontained no account of causes, no dialectical discrimination, and no 
(philosophic thought. 

These passages and two others in which Proclus refers to "the 

•commentators'" suggest that these commentators were numerous. 

He does not however give many names; and no doubt the only 

mportant commentaries were those of Heron, Porphyry, and Pappus. 

1 Proclus, p. 84, 8. 

' i.e. we retxih a certain height, use the platform so attained' as a base on which to build 
nother stage, then use that as a base and so on. 
* Proclus, p. 100, 10. * idid. p. 433, 15. ' iduf, p. 389, 11 ; p. 338, 16. 

2 — 2 



M INTRODUCTION [cK. ill 

I. Heron* 

Proclus alludes to Heron twice as Heron ntechaniois^, in another 
place* he associates him with Ctesibius. and in the three other 
passages* where Heron is mentioned there is no reason to doubt 
that the same person is meant, namely Heron of Alexandria. The 
date of Heron is still a vexed question, though the possible limits 
appear to have been practically narrowed down to the 150 years 
between (say) 50 B,c, and 100 A.U Martin* concluded that Heron 
lived till the middle of the first century B.C^ Hultsch* placed him at 
the end of the secoqd century B*C, Cantor in his first two editions 
took a middle course and gave 100 B.C as the date when he flourished*. 
But it is now certain that in his Mec/uinks, preserved in the Arabic 
and recently published ^ Heron quotes Posidonius the Stoic (of 
Apamea, Cicero's teacher) by name as the author of a definition 
of the centre of gravity. Now Posidonius lived till about the middle 
of the first century B.C. * and, assuming that his writings dated from 
not earlier than 90 or 80 B,C, we must put Heron at all events (say) 
fifty years later than Hultsch placed him. Cantor now, while main- 
taining that he belong^ to the first century B.C, admits that he may 
have flourished as late as the last third of it*. 

But in the meantime an entirely different view was elaborated by 
W* Schmidt, the editor of the first volume of the new edition of 
Heron's complete works, who assigned him to the second half of the 
first century a,d.* The arguments for the Urminus post gmm are 
mainly these* (i) Vitruvius gives in the preface to Book vii, of his 
De ArchiUctura (brought out apparently 14 B.C:) a list of authorities 
on machinatiofies from whom he had made extracts. This list contains 
twelve names and has every appearance of being scrupulously com- 
plete ; but^ while it includes Archytas, Archimedes, Ctesibius, and 
Fhilo of Byzantium (who come second, third, fourth, and sixth in 
order respectively), Heron is not mentioned. Moreover the pointii 
of diflfcrence between Vitruvius and Heron seem on the whole to be 
more numerous and important than the resemblances, (2) Diels 
concluded from the use of Latinisms by Heron that the first century a»D. 
was the earliest possible date. (3) A definite date was derived by 
Carra de Vaux from the identification of a small single-screw olive 
press described by Heron {Ah^c/mnicSj llh 20) with one mentioned by 
Pliny {Nat Hist XViil. 517) as having been introduced within the 
last twenty- two years : this gives A.D. 55 as the date before which the 
Mechanics could not have been written. The tenninus ante qnetn, 
100 A.D., was arrived at (i) from internal evidence suggesting that 

^ Proclus, p. 305, 24; p, 34^5, J3» 

* iMd. p, 41^ to. ' itid. p. 196, 16; p, 313, 71 p. 419, 13. 

* Martin, Reckfri/ui sur la xnt tt Its suvrngn d^Hfr&n d^AU^^mdrity Parish 1854, p* ^7, 
' Hultsch, Meir^hgum^m icriptorum reliqumtt 1&64, 1. 9. 

* Cantorf Gtuh. d, Math^ i^, p, 347. 
' Hsrm^is AUxandrini cpera quae tupertuni pmnia ^Teabner, Leipzig, vol* JL edited b* 

L* Nix and W- Schmidt, 1900. 

' Cantor, Gtsch. d. Matk. I,, p. 366. 

* See ff^rpmi AUxattdnm p/era, vol 1-, 1899, pp, bt— xxv. 



■ 



CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS ax 

Heron was earlier than Claudius Ptolemy (about 100-178 A.D.), and 
(2) from an apparent reference by Plutarch to a proposition about 
incidence and reflexion taking place at equal angles, proved by 
Heron in his Catoptricay coupled with the facts that in that work 
Heron mentions Menelaus of Alexandria (about 100 AD.) and that 
Plutarch died at a great age in 1 20 AD. 

Attempts have however been made in two recent tracts to over- 
throw almost the whole of these arguments ^ (i) It is asserted that the 
olive-press of Mechanics III. 20 is not the same as that referred to by 
Pliny. (2) It is pointed out that Heron is mentioned with Archimedes 
and Ctesibius in a passage of Proclus which is supposed to be drawn 
from Geminus*. But, as Geminus wrote about 70 B.C and Posidonius 
not earlier than 90 B.C., while Heron quotes Posidonius and is therefore 
later, the intervals are all too short to make it probable that Heron 
would be mentioned in Geminus' historical work ; and I think that 
the name of Heron may well have been inserted after that of Ctesibius 
by Proclus himself. (3) The view that Vitruvius did not use Heron's 
work is attacked, and the contrary sought to be proved, on the basis 
apparently of three passages, {a) Vitruvius* water-organ is held to 
be decidedly better than Heron's* : therefore Vitruvius used Heron's 
in order to improve upon it. {b) Vitruvius, in a passage describing 
a certain use of the lever, takes a wrong point to be the fulcrum ; and 
it is held that he cannot have made the mistake himself, but must 
necessarily have copied it from Heron^ In order, however, to find 
the same error in Heron, Hoppe arbitrarily alters both the figure and 
the text (c) Vitruvius describes the working of a certain crane in 
language less clear than that of Heron*; therefore he used Heron but 
misunderstood him ! AH would appear to be grist which comes to 
the mill of such critics : but I doubt whether such arguments will 
convince those who hold to the second half of the first century as the 
date that their view is mistaken. 

That Heron wrote a systematic commentary on the Elements 
might be inferred from Proclus, but it is rendered quite certain by 
references to the commentary in Arabian writers, and particularly in 
an-Nairizi's commentary on the first ten Books of the Elements. The 
Fihrist says, under Euclid, that " Heron wrote a commentary on this 
book [the Elements], endeavouring to solve its difficulties*"; and 
under Heron, " He wrote : the book of explanation of the obscurities 
in Euclid'...." An-Nairlzl's commentary quotes Heron by name very 
frequently, and often in such a way as to leave no doubt that the 
author had Heron's work actually before him. Thus the extracts are 

^ E. Hoppe, Ein Bdtrag tur Zeithatimmung Herons von AUxandrienf Hamburg, 1909 ; 
Rudolf Meier, De Heronis atUUe^ Leipzig, 1905. See the references to the arguments in 
Cantor, Gesch. d. Math, i„ pp. 365, 367, 545—7. 

' Proclus, p. 41, la 

' Vitruvius, x. 13 ; Heron, vol. i. p. 19a sqq. (PntuwuUus^ I. 43, 43). 

^ Vitruvius, X. 3, 3; Heron, vol. ii. pp. 114 — 116 (Mechanics^ 1 1. 8). 

■ Vitruvius, X. 3, 10 ; Heron, vol. ii. pp. 101 — 4 Qifechanus^ ill. a). 

* DcLs MaikimoHker- Veneuhniss im Fikrist (tr. Suter), p. 16. 

^ ibid, p. as. 



» INTRODUCTION [ch, hi 

given in the first person introduced by " Heron says *' (" Dixit Yrinus " 
or "Heron"); and in other places we are told that Heron '*says 
nothing/* or "is not found to have said anything " on such and such 
a proposition. The commentary of an-Nair!zi is being published by 
Besthorn and Heiberg from a Leiden MS. of the translation of the 
Elements by al-Hajjaj with the commentary attached \ But this MS. 
only contains six Books, and several pages in the first Book are 
missing> which contain the comments of Simphcius on the first twenty- 
two definitions of the first Book, Fortunately the commentary of 
an-Nairizi has been discovered in a more complete form, in a Latin 
translation by Gherardus Cremonensis of the twelfth century, which 
contains the missing comments by Simplicius and an-Nairizl's com- 
ments on the first ten Books. This valuable work has recently been 
edited by Curtzel 

Thus from the three sources, Prod us, and the two versions of 
an-NairlzI, which supplement one another, we are able to form a very 
good idea of the character of Heron's commentary. In some cases 
observations given by Proclus without the name of their author are 
seen from an-Nairlzl to be Heron's ; in a few cases notes attributed 
by Proclus to Heron are found in an-Nairizi without Heron's name ; 
and J curiously enough, one alternative proof (of L 25) given as Heron's 
by Proclus is introduced by the Arab with the remark that he has 
not been able to discover who is the author. 

Speaking generally, the comments of Heron do not seem to have 
contained much that can be called important We find 

(i) A few general notes, eg. that Heron would not admit more 
than three axioms. 

(2) Distinctions of a number of particular cases of Euclid's pro- 
positions according as the figure is drawn in one way or in another. 

Of this class are the different cases of L 35, 36, III, 7, 8 (where the 
chords to be compared are drawn on different sides of the diameter 
instead of on the same side), III. 12 (which is not Euclid's, but Heron's 
own, adding the case of external contact to that of internal contact in 
III, 11), VL 19 (where the triangle in which an additional line is drawn 
is taken to be the smaller of the two), vii, 19 (where he gives the 
particular case of three numbers in continued proportion^ instead of 
four proportionals). 

(3) Alternative proofs. Of these there should be mentioned {a) 
the proofs of II. i — 10 " without a figure/' being simply the algebraic 
forms of proof, easy but uninstructive, which are so popular nowadays^ 
the proof of 11 1, 25 (placed after 11 L 30 and starting from the arc 
instead of the chord), IIL 10 (proved by llh 9), m, 13 (a proof 
preceded by a lemma to the effect that a straight line cannot meet a 
circle in more than two points). Another class of alternative proof is 

^ CodAX Lddtmis 399, t, Euclidis Ekmmia ex iHt£rpr^iUi<mt ai-Hadstkdickndschii 
cum amrturUariii ai-Naruii. Two parts carrying the work to the end of Book I, were 
tssQed in 189^ and 1897 lespectively' Another part came out in 1905. 

* Anariitt in decern iibros £rwrts iiemeniarum Euciidu t&mmeniarii tx mUfprd^xUone 
Ghm^rdi Cremonensis.., aUdU Maxuniliaous Curtze (Teubner, Lelpdg^ 'S99)< 



I 



n 



CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS 23 

{b) that which is intended to meet a particular objection (h^tmuri^) 
which had been or might be raised to Euclid's construction. Thus 
in certain cases he avoids producing a particular straight line, where 
Euclid produces it, in order to meet the objection of any one who should 
deny our right to assume that there is any space available\ Of this 
class are Heron's proofs of I. 1 1, 1. 20, and his note on I. 16. Similarly 
on I. 48 he supposes the right-angled triangle which is constructed to 
be constructed on the same side of the common side as the given 
triangle is. A third class (c) is that which avoids reductio ad 
absurdum. Thus, instead of indirect proofs. Heron gives direct 

K roofs of I. 19 (for which he requires, and gives, a preliminary 
jmma), and of i. 25. 

(4) Heron supplies certain converses of Euclid's propositions, 
e.g. converses of li. 12, 13, vill. 27. 

(5) A few additions to, and extensions of, Euclid's propositions 
are also found. Some are unimportant, e.g. the construction of isosceles 
and scalene triangles in a note on I. i, the construction of two tangents 
in III. 17, the remark that Vll. 3 about finding the greatest common 
measure of three numbers can be applied to as many numbers as we 
please (as Euclid tacitly assumes in VII. 31). The most important 
extension is that of ill. 20 to the case where the angle at the 
circumference is greater than a right angle, and the direct deduction 
from this extension of the result of III. 22. Interesting also are the 
notes on I. 37 (on I. 24 in Proclus), where Heron proves that two 
triangles with two sides of one equal to two sides of the other and 
with the included angles supplementary are equal, and compares the 
areas where the sum of the two included angles (one being supposed 
greater than the other) is less or greater than two right angles, and 
on I. 47, where there is a proof (depending on preliminary lemmas) of 
the fact that, in the figure of the proposition, the straight lines AL^ 
BKy CF meet in a point. After iv. 16 there is a proof that, in a 
regular polygon with an even number of sides, the bisector of one 
angle also bisects its opposite, and an enunciation of the corresponding 
proposition for a regular polygon with an odd number of sides. 

Van Pesch" gives reason for attributing to Heron certain other 
notes found in Proclus, viz. that they are designed to meet the same 
sort of points as Heron had in view jn other notes undoubtedly written 
by him. These are (a) alternative proofs of I. 5, I. 17, and l. 32, 
which avoid the producing of certain straight lines, {p) an alternative 
proof of I. 9 avoiding the construction of the equilateral triangle on 
the side of BC opposite to A ; (^r) partial converses of I. 35 — 38, starting 
from the equality of the areas and the fact of the parallelograms or 
triangles being in the same parallels, and proving that the bases are 
the same or equal, may also be Heron's. Van Pesch further supposes 
that it was in Heron's commentary that the proof by Menelaus of 
L 25 and the proof by Philo of i. 8 were given. 

^ Cf. Prodns, 375, 7 c/ M X^oc rif rbww fi^ e^d^cu..., 389, 18 X^ci oSy nt 5rc oix Irrc 
^ . rinrot.,., 
c * De ProclifoniibuSt Lugduni-Batavorum, 1900. 



11 



\l 



24 INTRODUCTION [ch< hi 

The last reference to Heron made by an-Nairt^I occurs in the note 
on VI IL 27, so that the commentary of the former must at least have 
reached that point. 

II* Porphyry. 

The Porphyry here mentioned is of course the Neo-PIatonist who 
lived about 232-304 A.D. Whether he really wrote a systematic 
commentary on the £/anmfs is uncertain. The passages in Froclus 
which seem to make this probable are two in which he mentions him 
(i) as having demonstrated the necessity of the words "not on the 
same side" in the enunciation of L 14\ and (2) as having pointed out 
the necessity of understanding correctly the enunciation of L 26, since, 
if the particular injunctions as to the sides of the triangles to be taken 
as equal are not regarded, the student may easily fall into error'. , 
These passages, showing that Porphyry carefully analysed Euclid *s 
enunciations in these cases, certainly suggest that his remarks were 
part of a systematic commentary. Further, the list of mathematicians 
in the Fihrist gives Porphyry as having written '*a book on the 
Elements," It is true that Wenrich takes this book to have been a 
work by Porphyry mentioned by Suidas and Proclus ( Tfuohg. Platen, \ 
TT^pi apymv libri IL* 

There is nothing of importance in the notes attributed to Potphyry 
by Proclus. 

(i) Three alternative proofs of L 20, which avoid producing a side 
of the triangle, are assigned to Heron and Porphyry without saying 
which belonged to which. If the first of the three was Heron's, I 
agree with van Pesch that it is more probable that the two others 
were both Porphyry's than that the second was Heron's and only the 
third Porphyry's. For they are similar in character, and the third 
uses a result obtained in the second*. 

(2) Porphyry gave an alternative proof of L iS to meet a childish 
objection which is supposed to require the part of ^C equal to AB to 
be cut off from CA and not from A C. 

Proclus gives a pr^iisely similar alternative proof of I, 6 to meet a 
similar supposed objection ; and it may well be that, though Proclus 
mentions no name, this proof was also Porphyry's, as van P^ch 
suggests*. 

Two other references to Porpbyry found in Proclus cannot have 
anything to do with commentaries on the Elements. In the first a 
work called the Xu^fiiKTa is quoted, while in the second a philo- 
sophical question is raised, 

in. Pappus. 

The references to Pappus in Proclus arc not numerous ; but we 
have other evidence that he wrote a commentary on the Elements, 
Thus a scholiast on the definitions of the Data uses the phrase " as 

* Proclas, pp. 197, t— 1981 10. * iMd. p. 353, 13, 14 and ihe pages prtcedjng. 

* Fihrist (tr, Suter), p. 9^ lo and p> 45 {note j). 

* Van Pesch, Di Pr&cH fantihu^ pp. 119, 130* Heibcrg assigned Ibero as above in hts 
Eukiid-Studun (p, t6o)t but «eems to OAve changed Ms view later. (See Be$thoni*Heibetg, 
C^tM LiidfHjis^ p- 93, note 3.) 

* Van Pesch, ^* fit, pp. 13^— j. _ ^ 



l^ 



CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS 25 

Pappus says at the b^inning of his (commentary) on the loth (book) 
of Euclid*." Again in the Fikrist we are told that Pappus wrote a 
commentary to the tenth book of Euclid in two parts*. Fragments 
of this still survive in a MS. described by Woepcke*, Paris. No. 952. 2 
(supplement arabe de la Biblioth^que imp^riale), which contains a 
translation by Abu 'Uthman (beginning of loth century) of a Greek 
commentary on Book x. It is in two books, and there can now be 
no doubt that the author of the Greek commentary was Pappus*. 
Again Eutocius, in his note on Archimedes, On the Sphere and 
Cylinder!, 13, says that Pappus explained in his commentary on the 
Elements how to inscribe in a circle a polygon similar to a polygon 
inscribed in another circle ; and this would presumably come in his 
commentary on Book xil., just as the problem is solved in the second 
scholium on Eucl. xil. i. Thus Pappus' commentary on the Elements 
must have been pretty complete, an additional confirmation of this 
supposition being forthcoming in the reference of Marinus (a pupil 
and follower of Proclus) in his preface to the Data to "the com- 
mentaries of Pappus on the book'." 

The actual references to Pappus in Proclus are as follows: 
(i) On the Postulate (4) that all right angles are equal, Pappus is 
quoted as saying that the converse, viz. that all angles equal to a 
right angle are right, is not true*, since the angle included between 
the arcs of two semicircles which are equal, and have their diameters 
at right angles and terminating at one point, is equal to a right angle, 
but is not a right angle. 

(2) On the axioms Pappus is quoted as saying that, in addition to 
Euclid's axioms, others are on record as well (avpavaypaf^ecrOai) about 
unequals added to equals and equals added to unequals^; these, says 
Proclus, follow from the Euclidean axioms, while* others given by 
Pappus are involved by the definitions, namely those which assert 
that *' all parts of the plane and of the straight line coincide with one 
another," that " a point divides a straight line, a line a surface, and a 
surface a solid/' and that "the infinite is (obtained) in magnitudes 
both by addition and diminution'." 

^ Euclid's Da/a, ed. Menge, p. 363. ' Fikrist (tr. Suter), p. 29. 

• Mhnoires prheniis h tacacUmie des sciences^ 1856, XI v. pp. 658 — 710. 

* Woepcke read the name of the aathor, in the title of the first book zsB .los (the dot 
representing a missing vowel). He quotes tdso from other MSS. (e.g. of the Ta'riJkh ai- 
aukamU and of the Fikrist) where he reads the name of the commentator %& B ,lis^ B,n.s 
OT B./,s. Woepcke takes this author to be Vtdens, and thinks it possible that he may be 
the same as the astrologer Vettius Valens. This Heiberg {Euk/ia-Stsuiien, pp. 169, i7o) 
proves to be impossible, because, while one of the Mss. quoted by Woepcke says that 
**B,M.s, le Roimi'' (late-Greek) was later than Claudius Ptolemy and the Fikrist says 
**B./,s, \e RoUmi" wrote a commentary on Ptolemy's Plcmispkaerium, Vettius Valens 
seems to have lived under Hadrian, and must therefore have been an eider contemporary of 
Ptolemv. But Suter shows (Fikrist^ p. 22 and p. 54, note 9)) that Bancs is only distin- 
guished from Badcs by the position of a certain dot, and Ba/os may also easily have arisen 
from an original Bados (there is no P in Arabic), so that Pappus must be the person meant. 
This is further confirmed by the fact that the Fikrist gives this author and Valens as the 
subjects of two separate paragraphs, attributing to the latter astrological works only. 

" Heiberg, Euklid-Studien^ p. 173; Euclid's Daia^ ed. Mexige, pp. 356, Hi. 

* Proclus, pp. 189, 190. ' ibid. p. 197, (^10. 

• ihid, p. 198, 3—15. 




26 INTRODUCTION 

(3) Pappus gave a pretty proof of L 5. This proof has, I think; 
been wrongly understood ; on this point sec my note on the 
proposition. 

(4) On h 47 Proclus says^ : ** As the proof of the writer of the 
Elements is manifest, I think that it is not necessary to add anything 
further, but that what has been said is sufficient, since indeed those 
who have added more^ like Heron and Pappus, were obliged to make 
use of what is proved in the sixth book, without attaining any 
important result.'* We shall see what Heron's addition consisted of; 
what Pappus may have added we do not know, unless it was some- 
thing on the lines of his extension of L 47 found in the Syfu^ge 
(iv/p. 176, ed. Hultsch). 

We may fairly conclude, with van Pesch*, that Pappus is drawn 
upon in various other passages of Proclus where he quotes no 
authority, but where the subject-matter reminds us of other notes 
expressly assigned to Pappus or of what we otherwise know to have 
been favourite questions with him. Thus : 

r We are reminded of the curvilineal angle which is equal to but 
not a right angle by the note on I. 32 to the efllcct that the converse 
(that a figure with its interior angles together equal to two right 
angles is a triangle) is not true unless we confine ourselves to 
rectilineal figures. This statement is supported by reference to a 
figure formed by four semicircles whose diameters form a square, and 
one of which is turned inwards while the others are turned outwards. 
The figure forms two angles "equal to" right angles in the sense 
described by Pappus on Post 4, while the other curvilineal angles are 
not considered to be angles at all, and are left out in summing the 
internal angles. Similarly the allusions in the notes on L 4, 23 to 
curvilineal angles of which certain moon-shaped angles {p.i}PQ€i&€h) 
are shown to be "equal to*' rectilineal angles savour of Pappus, 

2, On L 9 Proclus says* that ** Others, starting from the Archi- 
medean spirals, divided any given rectilineal angle in any given ratio." 
We cannot but compare this with Pappus IV. p, 286, where the spiral 
is so used ; hence this note, including remarks immediately preceding 
about the conchoid and the quadratrix, which were used for the same 
purpose, may very well be due to Pappus. 

3, The subject of isopen metric figures was a favourite one with 
Pappus, who wrote a recension of Zenodorus' treatise on the subject* 
Now on L 35 Proclus speaks* about the paradox of parallelograms 
having equal area (between the same parallels) though the two sides 
between the parallels may be of any length, adding that of parallelo- 
grams with equal perimeter the rectangle is greatest if the base be 
given, and the square greatest if the base be not given etc. He 
returns to the subject on I. 37 about triangles* Compare' also his 
note on l. 4. These notes may have been taken from Pappus. 

* Proclus p, 419, g— '5- 

' Van Pcsch, Dt Procli f^rUibtt^^ y, 134 sqq. * Proclus, p. 171, jo* 

* Pappu^ V. pp. 304—^50; for Zenodorus own treatise see HuJ tech's Appendix, pp. it 89 
— i*n* 

* Produ% pp. 396 — 8. • iMd. pp. 403 — 4- '' iHd. pp. 336 — 7, t 



CH-Jii] GREEK COMMENTATORS OTHER THAN PROCLUS a? 

4. Again, on I. 21, Proclus remarks on the paradox that straight 
lines may be drawn from the base to a point within a triangle which 
are (i) together greater than the two sides, and (2) include a less 
angle provided that the straight lines may be drawn from points in 
the base other than its extremities. The subject of straight lines 
satisfying condition (i) was treated at length, with reference to a 
variety of cases, by Pappus*, after a collection of "paradoxes" by 
Erycinus, of whom nothing more is known. Proclus gives Pappus' 
first case, and adds a rather useless proof of the possibility of drawing 
straight lines satisfying condition (2) alone, adding that " the proposi- 
tion stated has been proved by me without using the parallels of 
the commentators'." By " the commentators " Pappus is doubtless 
meant 

5. Lastly, the "four-sided triangle," called by Zenodorus the 
••hollow-angled,"* is mentioned in the notes on I. Def. 24 — 29 and 
I. 21. As Pappus wrote on Zenodorus' work in which the term 
occurred^ Pappus may be responsible for these notes. 

IV. Simplicius. 

According to the Fihrist^, Simplicius the Greek wrote ''a com- 
mentary to the beginning of Euclid's book, which forms an introduc- 
tion to geometry." And in fact this commentary on the definitions, 
postulates and axioms (including the postulate known as the Parallel- 
Axiom) is preserved in the Arabic commentary of an-Nairizi*. On 
two subjects this commentary of Simplicius quotes a certain " Aganis," 
the first subject being the definition of an angle, and the second the 
definition of parallels and the parallel-postulate. Simplicius gives 
word for word, in a long passage placed by an-NairlzI after I. 29, an 
attempt by •• Aganis " to prove the parallel-postulate. It starts from 
a definition of parallels which agrees with Geminus' view of them as 
given by Proclus^ and is closely connected with the definition given 
by Posidonius*. Hence it has been assumed that "Aganis" is none 
other than Geminus, and the historical importance of the commentary 
of Simplicius has been judged accordingly. But it has been recently 
shown by Tannery that the identification of " Aganis " with Geminus 
is practically impossible*. In the translation of Besthom-Heiberg 
Aganis is called by Simplicius in one place " philosophus Aganis," in 
another " magister noster Aganis," in Gherard's version he is " socius 
Aganis" and "socius noster Aganis." These expressions seem to 
leave no doubt that Aganis was a contemporary and friend, if not 
master, of Simplicius ; and it is impossible to suppose that Simplicius 
(fl. about 500 A.D.) could have used them of a man who lived four and 

• Pappus III. jM). 104—130. » Proclus, p. 318, 15. 

' Proclus, p. 105, 14; cf. pp. 318, 339. * See Pappus, ed. Hultsch, pp. 11 54, 1306. 

• Fihrist (tr. Suter), p. ii. 

• An-NairizI, cd. Bestbom-Heiberg, pp. 9—41, 119;— 1331 cd. Curtze, pp. i — 37, 65 — 73. 
The Codix LHdensiSy from which ^sthom and Heiberg are editing the work, has un- 
fortunately lost some leaves so that there is a gap from Def. i to Def. 35 (parallels). The 
loss is, however, made good by Curtze's edition of the translation by Ghenmi of Cremona. 

' Proclus, p. 177, 31. • ibid, p. 176, 7. 

' Biblioikeca Mathematical ii^, 1900, pp. 9 — 11. 



"1 

tS INTRODUCTION [ciL m J 

a half centuries before his time. A phrase in Simplicius* word-for- 
word quotation from Aganis leads to the same conclusion. He speaks 
of people who objected *'cven in ancient times" (iam antiquitus) to ^ 
the use by geometers of this postulate. This would not have been an | 

appropriate phrase had Gcminus been the writer I do not think 
that this difficulty can be got over by Suter's suggestion* that the 
passages in question may have been taken out of Heron's commentary, 
and that an-Nairizi may have forgotten to name the author ; it seems 
clear that Simplicius is the person who described " Aganis." Hence 
we are driven to suppose that Aganis was not Geminus, but some 
unknown contemporary of Simplicius*, Considerable interest will 
however continue to attach to the comments of Simplicius so 
fortunately preserved* 

Proclus tells us that one Aegaeas {? Aenaeas) of Hierapolis wrote an 
epitome of the Ekments^ ; but we know nothing more of him or of it 

1 Z€its€krififur Ai^th. u. Pkysik, XLIV., hLit.-tttt Abtb Jf. 6j* 

^ The ibove Aripimeni seems Lome quUc infiuperahte. Ine other Ai^inents of Tvunery 

do Dot* however, carry conviction to my mind* I dci ttot follow the reaM>niiig bailed on 
Aganis' definition of an angle. It appears to me a pure assumption ihat Geminus would have 
seen that Po^idonius^ definition of pftralleb w^ not admissible. Nor 6ot^ it seem to me to 
count for mych that Proclus, while telling us that Geminus held that the postulate ought to be 
proved and warned the unwaiy again^ hastUy concluding that two straight lines approaching 
one another mu&t necci^sarity meet (cf« a curve and it& asymptote), gives no nint that 
Geminus did try to prove the postulate^ It may well be tnat Procltis omitted Getninns* 
"proofs (if he trrote one) because he preferred Ptolemy*s attempt which he give 

(PP- 3^5—7)' 

* Proclus, p. 361* 11* 



CHAPTER IV. 

PROCLUS AND HIS SOURCES^. 

It is well known that the commentary of Proclus on Eucl. Book I. 
is one of the two main sources of information as to the history of 
Greek geometry which we possess, the other being the Collection of 
Pappus. They are the more precious because the original works of 
the forerunners of Euclid, Archimedes and Apollonius are lost, having 
probably been discarded and forgotten almost immediately after the 
appearance of the masterpieces of that great trio. 

Proclus himself lived 410-485 A.D., so that there had already 
passed a sufficient amount of time for the tradition relating to the 
pre-Euclidean geometers to become obscure and defective. In this 
connexion a passage is quoted from Simplicius' who, in his account 
of the quadrature of certain lunes by Hippocrates of Chios, while 
mentioning two authorities for his statements, Alexander Aphro- 
disiensis (about 220 A.D.) and Eudemus, says in one place', ''As 
regards Hippocrates of Chios we must pay more attention to Eudemus, 
sittce he was nearer the times, being a pupil of Aristotle." 

The importance therefore of a critical examination of Proclus' 
commentary with a view to determining from what original sources 
he drew need not be further emphasised. 

Proclus received his early training in Alexandria, where Olympio- 
dorus was his instructor in the works of Aristotle, and mathematics 
was taught him by one Heron* (of course a different Heron from the 
'^mechanicus Hero" of whom we have already spoken). He after- 
wards went to Athens where he was imbued by Plutarch, and by 
Syrianus, with the Neo-Platonic philosophy, to which he then devoted 

^ My task in this chapter is made easy by the appearance, in the nick of time, of the 
dissertation De Prodi fotUitms by J. G. van Pesch (Lugduni-Batavorum, Apud L. van 
Nifterik, MDCCCC). The chapters dealing directly with the subject show a thorough 
acquaintance on the part of the author with all the literature bearing on it; he covers 
the whole field and he exercises a sound and sober judgment in forming his conclusions. 
The same cannot always be said of his only predecessor in the same inquiry, Tannery 
(in La GhmHrie grtcque, 1887), ^^^ often robs his speculations of much of their value 
through his proneness to run away with an idea ; he does so in this case, basing most of his 
conclusions on an arbitrary and unwarranted assumption as to the significance of the words 
ol repf rua (e.g.'Upfura, no^ctdt&rtov etc) as used in Proclus. 

• Simplicius on Aristotle*s Physics^ ed. Dieb, pp. 54 — 69. 

* ibid, p. 6%, 31. 

^ Cf. Martin, Recktrcha sur lavidet Us omfrages d^Hiron dAUxandrii, pp. 340 — 3. 



30 INTRODUCTION [ch. iv 

heart and soul, becoming one of its most prominent exponents* He 
speaks everywhere with the highest respect of his masters, and 
was in turn regarded with extravagant veneration by his contem- 
poraries, as we learn from Marinus his pupil and biographer. On 
the death of Syrian us he was put at the head of the Neo-Platonic 
school He was a man of untiring industry^ as is shown by the 
number of books which he WTote, including a large number of com- 
mentaries, mostly on the dialogues of Plato* He was an acute 
dialectician, and pre- eminent among his contemporaries in the 
range of his learning^; he was a competent mathematician; he was 
even a poet At the same time he was a believer in all sorts of 
myths and mysteries and a devout worshipper of divinities both 
Greek and Oriental, 

Though he was a competent mathematician, he was evidently 
4nuch more a philosopher than a mathematician I This is shown 
even in his commentary on EucL !.» where, not only in the Prologues 
(especially the first^ but also in the notes themselves, he seizes any 
opportunity for a philosophical digression. He says himself that he 
attaches most importance to **the things which require deeper study 
and contribute to the sum of philosophy*"; alternative proofs, cases ^ 
and the like (though he gives many) have no attraction for him ; 
and, in particular, he attaches no value to the addition of Heron to 
L 47*, which is of considerable mathematical interest Though he 
esteemed mathematics highly, it was only as a handmaid to philosophy. 
He quotes Plato's opinion to the effect that "mathematics, as making 
use of hypotheses, falls short of the non-hypothetical and perfect 
science*". »/' Let us then not say that Plato excludes mathematics 
from the sciences, but that he declares it to be secondary to the one 
supreme science V And again, while "mathematical science must be 
considered desirable in itself, though not with reference to the needs 
of daily lifej'' **if it is necessary to refer the benefit arising from it to 
, something else, we must connect that benefit with intellectual know- 
ledge {voepcLv yptii>a-tp)y to which it leads the way and is a propaedeutic, 
clearing the eye of the soul and takii)g away the impediments which 
the senses place in the way of the knowledge of universals {twv 
Skmvyr 

We know that in the Neo* PI atonic school the younger pupils 
learnt mathematics ; and it is clear that Proclus taught this subject, 
and that this was the origin of the commentary. Many passages 
show him as a master speaking to scholars. Thus "we have illustrated 

^ Edict c^ Is him '* Der GelchrtCi dem kein Feld damaligen Wissens verschlos^cn ist/^ 
' Van Tesch observes that in his commentaries on the Tiwa^iw (pp* 67;— 1> he fipe^ks 
as no real maihemaiician could h^ve spoken. In the passage referred Lo ihe qu^uon h 
whether the &un occupies a middie place among the planets^ Proclus rejects the view of 
Hipparchus and Ptolemy because **d &toupy6v^^ ($c. the Chaldean, says Zellcr) thinks otherwise , 
"whom it is not lawful to disbeJieve/' Martin says rather neatly^ *^ Pour ProduSi les 
Elements d'EucUde ont Thcnrcu^e chance de n'^tre contr«dits ni par fei Omclcs cbaldaSfques, 
tu par tes speculations des pythtigoneicns ancicns et nouveaux......" 

• Proclus, p* S41 13* ' iHd. p, 419^ Up 

* ibid^ p. 3t, 30» • ibid, p. %t, %" i 
^ itid. p. a;, a; to aS, 7; cf. alio p. 11, 95, pp» 46, 47. 



CH. TV] PROCLUS AND HIS SOURCES 31 



and made plain all these things in the case of the first problem, but 
it is necessary that my hearers should make the same inquiry as 
regards the others as welP," and '' I do not indicate these things as a 
merely incidental matter but as preparing us beforehand for the 
doctrine of the Timaeusl" Further, the pupils whom he was 
I • addressing were beginners in mathematics ; for in one place he says 
^ that he omits " for the present" to speak of the discoveries of those 
I who employed the curves of Nicomedes and Hippias for trisecting 
I an angle, and of those who used the Archimedean spiral for dividing 
an angle in any given ratio, because these things would be too 
difficult for beginners (SvaOeapiJTovi roU elaayofiepoi^y. Again, if 
his pupils had not been beginners, it would not have been necessary 
for Proclus to explain what is meant by saying that sides subtend 
certain angles^ the difference between adjacent and vertical angles' 
etc., or to exhort them, as he often does, to work out other particular 
cases for themselves, for practice {yviiva^ria^ ev€Ka)\ 

The commentary seems then to have been founded on Proclus' 
lectures to beginners in mathematics. But there are signs that it 
was revised and re-edited for a larger public ; thus he gives notice in 
one place' "to those who shall come upon" his work (to*9 ivrev^o- 
fihoi^y There are also passages which could not have been under- 
stood by the beginners to whom he lectured, e.g. passages about the 
cylindrical helix', conchoids and cissoids'. These passages may have 
been added in the revised edition, or, as van Pesch conjectures, the 
explanations given in the lectures may have been much fuller and 
more comprehensible to beginners, and they may have been shortened 
on revision. 

In his comments on the propositions of Euclid, Proclus generally 
proceeds in this way : first he gives explanations r^arding Euclid's 
proofs, secondly he gives a few different cases, mainly for the sake of 
practice, and thirdly he addresses himself to refuting objections 
raised by cavillers to particular propositions. The latter class of 
note he deems necessary because of "sophistical cavils" and the 
attitude of the people who rejoiced in finding paralogisms and in 
causing annoyance to scientific men". His commentary does not 
seem to have been written for the purpose of correcting or improving 
Euclid. For there are very few passages of mathematical content 
in which Proclus can be supposed to be propounding anythirtg of his 
own ; nearly all are taken from the works of others, mostly earlier 
commentators, so that, for the purpose of improving on or correcting 
Euclid, there was no need for his commentary at all. Indeed only in 
one place does he definitely bring forward anything of his own to get 
over a difficulty which he finds in Euclid"; this is where he tries to 

• Proclus, p. a 10, 18. • iMd, p. 384, 1. 

• Mi, p. 171, II. * Md. p. 138, 13. 

• Md. p. 198, 14. • Cf. p. 314, 15 (on 1. 1). 
' iHd. p. 84, g. • idtd, p. 105. 

9 .•A.W ^ .7^ 10 .A.V/ n -.fe ft 



wvut, ^* 04, y* 

' Md. p. 113. 

" Md. pp. 368—373. 



*• lAfV/. p. 375, 8. 



i- 



3a INTRODUCTION [ch. iv 

prove the parallel-postulate, after first giving Ptolemy's attempt and 
then pointing out objections to it. On the other hand there are a 
number of passages in which he extols Euclid ; thrice* also he 
supports Euclid against Apollonius where the laUer had given proofs 
which he considered better than Euclid's (I. lo, ii, and 23). 

Allusion must be made to the debated question whether Proclus 
continued his commentaries beyond Book L His intention to do so 
is clear from the following passages. Just after the words above 
quQted about the trisection etc* of an angle by means of certain curves 
he says, '* For we may perhaps more appropriately examine these 
things on the third book, where the writer of the Elements bisects a 
given circumference^" Again, after saying that of all paraJlelograms 
which have the same perimeter the square is the greatest *' and the 
rhomboid least of all/' he adds: "But this we wilt prove in another 
place; for it is more appropriate to the (discussion of the) hypotheses 
of the second book* " Lastly, when alluding (on h 45) to the squaring 
of the circle, and to Archimedes' proposition that any circle is equal 
to the right-angled triangle in which the perpendicular is equal to the 
radius of the circle and the base to its perimeter, he adds, '' But of this 
elsewhere**'; this may imply an intention to treat of the subject on 
Eucl. XIL, though Heiberg doubts it** But it is clear that, at the time 
when the commentary on Book I. was written* Proclus had not yet 
b^^n to write on the other Books and was uncertain whether he 
would be able to do so : for at the end he says\ '* For my part, if I 
should be able to discuss the other books^ in the same manner, I 
should give thanks to the gods ; but, if other cares should draw me 
away, I beg those who are attracted by this subject to complete the 
exposition of the other books as well, following the same method, and 
addressing themselves throughout to the deeper and better denned 
questions involved '' (to 'Trpayfjt^aretwS^^ vapra^ov xal ev&iaiperov 
fLeraStoi Kovras:)- 

There is in fact no satisfactory evidence that Proclus did actually 
write any more commentaries than that on Book J*, those who have 
attributed to Proclus some of the scholia on the later books having 
failed to prove their case\ The contrary view receives support from 
two facts pointed out by Heiberg, viz. (i) that the scholiast's copy 
of Proclus was not much better than our MSS. : in particular, it had 

* Proclus, p. aSo, gj p. 183* 10; pp, 335, $36, * i6id. p» 371, 14* 
■ i^J. p. J98, 1 8, * ihui, p, 433, 6. 

• Heiberg J £uklid*Studi£n, p* 165, note. ' Proclus, p. +33* 9. 

^ The words in ihe Greek arc: ti ptHv Sifwi^Btlnifiet x^i roTt XiHiro7» m ovr^ rpi/wof 
^\$w. For i^t\9ttif Heiberg would read iwf^fMtty. 

' Heiberg {EuMUd-Sindkn, p- 166) gives reason for doubling the evidoice mdduoed 
by Waclumutht by which Knoche vras persuaded to give up his ori^nal view (ha.t Proclus did 
nai write any more comtnentaries. Wacbimutb tie lies sole-ly upon a Vatican MS- which ha>i 
at the head of a collection of scholia on Books K (extracts from the e sclent convmcnUry 
of Proclus), IL, Vm VI., 9C. the title ; Eti ri EujfXff^n; ^rwx^ia wpo\aft^iu/6fiiPa in tQv Iip6ii\o*t 
€wopdhfw jfol jcat' iwiT^^. Hcibeig holds Umt this liile its^Jf makes ti probable that the 
anthor^ip ascribed to Proclus was restricted to the scholia on Book J.: otherwise how 
could one understand the expression irpoXati^ti^btitwa in tmp llp6K\ov^ which words would 
suit extracts ^m Frndtu'/fW^s^tfi weU enough, but not the acholia to latei^ Books? 



CH. nr] PROCLUS AND HIS SOURCES 33 

the same lacunae in the notes on l. 36, 37 and 1. 41 — ^43 : and even this 
fact makes it improbable that the scholiast had further commentaries 
of Proclus which have vanished for us ; (2) that there is no trace 
in the scholia of the notes which Proclus promised in the passages 
quoted above \ 

Coming now to the question of the sources of Proclus, we may say 
that everything goes to show that his commentary is a compilation, 
though a compilation "in the better sense" of the term*. He does 
not even give us to understand that we shall find in it much of his own ; 
" let us," he says, " now turn to the exposition of the theorems proved 
by Euclid, selecting the more subtle of the comments made on them 
by the ancient writers, and cutting down their interminable difluse- 
ness...*'': not a word about anything of his own. At the same time, 
he seems to imply that he will not necessarily on each occasion quote 
the source of each extract from an earlier commentary ; and, in fact, 
while he quotes the name of his authority in many places, especially 
where the subject is important, in many others, where it is equally 
certain that he is not giving anything of his own, he mentions no 
authority. Thus he quotes Heron by name six times ; but we now 
know, from the commentary of an-NairlzI, that a number of other 
passages, where he mentions no name, are taken from Heron, and 
among them the not unimportant addition of an alternative proof to 
I. 19. Hence we can by no means conclude that, where no authority 
is mentioned, Proclus is giving notes of his own. The presumption is 
generally the other way ; and it is often possible to arrive at a con- 
clusion, either that a particular note is not Proclus' own, or that it 
is definitely attributable to someone else, by applying the ordinary 
principles of criticism. Thus, where the note shows an unmistakable 
affinity to another which Proclus definitely attributes to some com- 
mentator by name, especially when both contain some peculiar and 
distinctive idea, we cannot have much doubt in assigning both to the 
same commentator^ Again, van Pesch finds a criterion in the form 
of a note, where the explanation is so condensed as to be only just 
intelligible ; the note is that in which a converse of I. 32 is proved*, 
the proposition namely that a rectilineal figure which has all its 
interior angles together equal to two right angles is a triangle. 

It is not safe to attribute a passage to Proclus himself because he 
uses the first person in such expressions as " I say " or " I will prove '* 
— for he was in the habit of putting into his own words the substance 
of notes borrowed from others — nor because, in speaking of an 

1 Heiberg, Euklid'Studim, pp. 167, 168. 

' Kooche, UfUersuchungen iiber des Froklus Diadochus Comnuntar tu Euklitts EU- 
menien (1863) p. 11. 

■ Proclus, p. aoo, 10 — 15. 

* Instances of the application of this criterion will be found in the discussion of Proclus* 
indebtedness to the commentaries of Heron, Porphyry and Pappus. 

* Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the 
proof is followed by a passage which, on comparison with Pappu? note on the postulate that 
all right angles are equal, he feeb justified in assigning to Pappus. I doubt if ttie evidence is 
sufficient. 

H. K. 3 



U INTRODUCTION [cH. nr 

objection raised to a particular proposition, he uses such expressions 
as "perhaps someone may object" (fir<w? S* av rtj^cT ipirraUp^^*): for 
sometimes other words in the same passage indicate that the objection 
had actually been taken by someone'. Speaking generally, we shall 
not be justified in concluding that Proclus is stating something new of 
his own unless he indicates this himself in express terms. 

As regards the form of Proclus* references to others by name, van 
Pesch notes that he very seldom mentions the particular wark from 
which he is borrowing. If we leave out of account the references to 
Plato's dialogues, there are only the following references to books : 
tiieBac^ia^ of Philolaus\ the Symmikta of Porphyry", Archimedes Oh 
the Sphere and Cyiindtr^^ Apollonius On the cochiias^^ a book by 
Eudemus on The Ang'ie\ a whole book of Posidonius directed against 
Zeno of the Epicurean sect^ Carpus* Astrofwmy^, Eudemus' History of 
Geometry^ ^ and a tract by Ptolemy on the parallel -postulate". 

Again, Proclus does not always indicate that he is quoting some- 
thing at second-hand. He often does so, e.g. he quotes Heron as the 
authority for a statement about PhiJippus, Eudemus as attributing a 
certain theorem to Oenopides etc.; but he says on h \2 that "Oeno- 
pides first investigated this problem, thinking it useful for astronomy** . 
when he cannot have had Oenopides* work before him. i 

It has been said above that Proclus was in the habit of stating in * 
his own words the substance of the things which he borrowed. We j 
are prepared for this when we find him stating that he will select the 
best things from ancient commentaries and "cut short their intermin- 
able diffuseness/* that he will * briefiy describe '* {avvrofi^K; itrTopTJaai) 
the other proofs of I. 20 given by Heron and Porphyry and also the 
proofs of I, 25 by M^nelaus and Heron, But the best evidence is of 
course to be found in the passages where he quotes works still extant, 
e*g. those of Plato, Aristotle and Plotinus, Examination of these 
passages shows great divergences from the original ; even where he 
purports to quote textual ly, using the expressions *' Plato saySj'' or 
•' Plotinus says," he by no means quotes word for word". In fact, he 
seems to have had a positive distaste for quoting textually from other 
works. He cannot conquer this even when quoting from Euclid ; he 
says in his note on I. 22, "we will follow the words of the geometer*' 
but fails, nevertheless, to reproduce the text of Euclid unchanged". 

We now come to the sources themselves from which Proclus drew 

^ Van Pesch illustrates this by an objection refuted in the note on i. 9, p. 173, ir sqq. 
After using the above expression to introduce the objection, Proclus uses further on (p. 373, 15) 
the term **they say" {^aaU^). 

» Proclus, p. 11, 15. ' idiit. p. 56, 15. 

* Mt.p. 71, 18. • tiiJ, p. J05, 5. 

* idut. p. 135, 8. ' s'h'd, p. 300, 3. 

* sHd, p. 341, 19. • i^ut. p. 353, 15. 
" Md. p. 363, 15. 
" See the passages referred to by van Pesch (p. 70). The most glaring case is a passage 

(p. 31, 19) where he quotes Plotinus, using the expression *' Plotinus says " Comparison 

with Plotinus, Ennead, I. 3. 3i shows that very few words are those of Plotinus himself; the 
rest represent Plotinus* views in Proclus' own language. 
" Proclus, p. 330, 19 sqq. 



r 



CH. iv] PROCLUS AND HIS SOURCES 35 

in writing his commentary. Three have already been disposed of, 
viz. Heron, Porphyry and Pappus, who had all written commentaries 
on the Elements^, We go on to 

Eudemus, the pupil of Aristotle, who, among other works, wrote a 
history of arithmetic, a history of astronomy, and a history of geometry. 
The importance of the last mentioned work is attested by the frequent 
use made of it by ancient writers. That there was no other history 
of geometry written after the time of Eudemus seems to be proved by 
the remark of Proclus in the course of his famous summary : " Those 
who compiled histories bring the development of this science up to 
this point Not much younger than tliese is Euclid*...." The loss of 
Eudemus' history is one of the gravest which fate has inflicted upon 
us, for it cannot be doubted that Eudemus had before him a number 
of the actual works of earlier geometers, which, as before observed, 
seem to have vanished completely when they were superseded by the 
treatises of Euclid, Archimedes and Apollonius. As it is, we have to 
be thankful for the fragments from Eudemus which such writers as 
Proclus have preserved to us. 

1 agree with van Pesch* that there is no sufficient reason for 
doubting that the work of Eudemus was accessible to Proclus at first 
hand. For the later writers Simplicius and Eutocius refer to it in 
terms such as leave no room for doubt that thty had it before them. 
I have already quoted a passage from Simplicius' account of the lunes 
of Hippocrates to the effect that Eudemus must be considered the 
best authority since he lived nearer the times^ In the same place 
Simplicius says', ** I will set out what Eudemus says word for word 
(xariL Xi^ip Xeyofi^va), adding only a little explanation in the shape of 
reference to Euclid's Elements owing to the memorandum-like style of 
Eudemus {h^a top vwofipnfiaTMOP rpoirop rov EvSif/xou) who sets out 
his explanations in the abbreviated form usual with ancient writers. 
Now in the second book of the history of geometry he writes as 
follows'." It is not possible to suppose that Simplicius would have 
written in this way about the style of Eudemus if he had merely been 
copying certain passages second-hand out of some other author and 
had not the original work itself to refer to. In like manner, Eutocius 
speaks of the paralogisms handed down in connexion with the 
attempts of Hippocrates and Antiphon to square the circle', "with 
which I imagine that those are accurately acquainted who have 
examined {iirecKefifiipov^) the geometrical history of Eudemus and 
know the Ceria Aristotelica." How could the contemporaries of Euto- 
cius have examified the work of Eudemus unless it was still extant in 
his time ? 

The passages in which Proclus quotes Eudemus by name as his 
authority are as follows : 

(i) On I. 26 he says that Eudemus in his history of geometry 

^ See pp. 10 to 17 above. 

• Proclus, p. 68, 4 — 7. ' De Proclifontibus, pp. 73 — 75. 

^ See above» p. 19. ^ Simplicius, loc. cit., ed. Diels, p. 60, 17. 

• Archimedes, ed. Heiberg, vol. in. p. 164. 



36 INTRODUCTION [ch, iv 

referred this theorem to Thales, inasmuch as it was necessary to 
Thales method of ascertaining the distance of ships from the shore', 

(2) Eudemus attributed to Thales the discovery of Eucl L 15', 
and 

(3) to Oenopides the problem of 1. 23". 

(4) Eudemus referred the discovery of the theorem in L 32 to the 
Pythagoreans, and gave their proof of it^ which Proclus reproduces*. 

(j) On I, 44 Proclus tells us* that Eudemus says that *' these 
things are ancient, being discoveries of the Pythagorean muse, the 
application (wapa^oXi^) of areas, their exceeding {yirfpffoXf)) and 
their falling short (eXXci^K)*" The next words about the appro- 
priation of these terms (parabola, hyperbola and ellipse) by later 
writers (i.e, Apollonius) to denote the conic sections are of course not 
due to Eudemus. 

Coming now to notes where Eudemus is not named by Proclus, 
we may fairly conjecture, with van Pesch, that Eudemus was really 
the authority for the statements (i) that Thales first proved that a 
circle is bisected by its diameter" (though the proof by nduciio ad 
absurdum which follows in Proclus cannot be attributed to Thales^), 
(2) that '* Plato made over to Leodamas the analytical method, by 
means of which it is recorded (/txTopi^ra*) that the latter too made 
many discoveries in geometry V* (3) that the theorem of I* 5 was due 
to Thales, and that for equal angles he used the more archaic 
expression "similar'' angles*, (4) that Oenopides first investigated 
the problem of L 12. and that he called the perpendicular the 
gfiomonic line {Kara yva^fioi'ay^, (5) that the theorem that only three 
sorts of polygons can fill up the space round a point, viz. the 
equilateral triangle, the square and the regular hexagon, was 
Pythagorean", Eudemus may also be the authority for Proclus' 
description of the two methods, referred to Plato and Pythagoras 
respectively, of forming right-angled triangles in w^hole numbers". 

We cannot attribute to Eudemus the beginning of the note on 
I, 47 where Proclus says that ''if we listen to those who like to 
recount ancient history, we may find some of them referring this 
theorem to Pythagoras and saying that he sacrificed an ox in honour 
of his discovery"/' As such a sacrifice was contrary to the Pytha- 
gorean tenets, and Eudemus could not have been unaware of this, 
the story cannot rest on his authority. Moreover Proclus speaks as 
though he were not certain of the correctness of the tradition ; indeed, 

* FrocliiS, p. 351, 14^ — 18. * i&iJ^ p. 399, 3. 

» iSit/. p* 333, 5. * iAid. p. 379, i— 16. 

* iM, p, 419, 15—18. * i^id, p. ij7, 10, It. 

^ Cvitor (^^fi. d. Alaih, ij, p. %%%) poinis Qui Uic connexion between the nducii^ ad 
ahiurdttm and the analyticiit method said to have been discovered by Piato. Proclus gives 
the proof by reductio ^ abiurdum to meet an imagiitary critic who d^tres a maihtmatkal 
proof ; possibly Thales may have been satisRei! with the argument in the same aeotence 
which mentions Thales, ^Mhe cause of the bisection being the unswerving course of the 
straight line through the centre." 

* Proclus, p. 31I9 19—33. ' i^' p. 350, 30. 

*• Md, p. S83, 7 — 10. " iHd. pp. 304, II — 305, 3. 

^ iHd. pp. 438, 7— 439, 9. ^ Hid. p. 4s6, 6—9. 



1 



\ 



r 



! 



CH. IV] PROCLUS AND HIS SOURCES 37 

so far as the story of the sacrifice is concerned, the same thing is told 
of Thales in connexion with his discovery that the angle in a semi- 
circle is a right angle*, and Plutarch is not certain whether the ox 
was sacrificed on the discovery of I. 47 or of the problem about 
application of areas*. Plutarch's doubt suggests that he knew of no 
evidence for the story beyond the vague allusion in the distich of 
Apollodorus "Logisticus" (the "calculator") cited by Diogenes 
Laertius also*; and Proclus may have had in mind this couplet with 
the passages of Plutarch. 

We come now to the question of the famous historical summary 
given by Proclus*. No one appears to maintain that Eudemus is the 
author of even the early part of this summary in the form in which 
Proclus gives it It is, as is well known, divided into two distinct 
parts, between which comes the remark, "Those who compiled 
histories' bring the development of this science up to this point. 
Not much younger than these is Euclid, who put together the 
Elements, collecting many of the theorems of Eudoxus, perfecting 
many others by Theaetetus, and bringing to irrefragable demonstration 
the things which had only been somewhat loosely proved by his pre- 
decessors." Since Euclid was later than Eudemus, it is impossible that 
Eudemus can have written this. Yet the style of the summary after 
this point does not show any such change from that of the former 
portion as to suggest different authorship. The author of the earlier 
portion recurs frequently to the question of the origin of the 
elements of geometry in a way in which no one would be likely to 
do who was not later than Euclid ; and it must be the same hand 
which in the second portion connects Euclid's Elements with the 
work of Eudoxus and Theaetetus'. 

If then the summary is the work of one author, and that author 
not Eudemus, who is it likely to have been ? Tannery answers that 
it is Geminus' ; but I think, with van Pesch, that he has failed to 
show why it should be Geminus rather than another. And certainly 
the extracts which we have from Geminus' work suggest that the sort 
of topics which it dealt with was quite different ; they seem rather to 
have been general questions of the content of mathematics, and even 
Tannery admits that historical details could only have come inci- 
dentally into the work'. 

Could the author have been Proclus himself.^ Circumstances 

1 Diogenes Laertius, i. 94, p. 6, ed. Cobet. 

' Plutarch, non posse suaviter vivi secundum Epicurum, 1 1 ; Symp, vili, i. 

' Diog. LAert vill. 11, p. 107, ed. Cobet: 

'HrUa Iiv$ay6f^ rb ve/xjcXe^r c0pero ypdfAfia^ 
K^uf 4^ Ih-ti) JcXeirV ^*7* fiovOvairfp. 
See on this subject Tannery, La CiomJtrie grecque^ p. 105. 

• Proclus, pp. 64 — 70. 

• The plural is well explained by Tannery, La GiomHrie grecque, pp. 73, 74. No doubt 
the author of the summary tried to supplement Eudemus by means of any other histories 
which threw light on the subject. Thus e.g. the allusion (p. 64, 11) to the Nile recalls 
Herodotus. Cf. the expression in Proclus, p. 64, 19, vapii tup roXXQv irr^ptfrak, 

• Tannery, La Gioniitrie grecque^ p. 75. 

' ibid, pp. 66—75. • »W. p. 19. 



» 



38 INTRODUCTION 

which seem to suggest this possibility are (i^ that, as already stated, | 

the question of the origin of the Ekmiuts is kept prominent, , 

(2) that there is no mention of Democritus, whom Eudemus would 
not be likely to have ignored, while a follower of Plato would be 
likely enough to do him the injustice, following the example of Plato 
who was an opponent of Democntus, never once mentions him, and 
is said to have wished to burn all his writings ^ and (3) the allusion at 
the beginning to the " inspired Aristotle ** (o tat^jiovto^ 'ApiaTOT€Xf}<;)\ 
though this may easily have been Inserted by Proclus tn a quotation 
made by him from someone else On the other hand there arc 
considerations which suggest that Proclus himself was mt the writer, 
(i) The style of the whole passage is not such as to point to him { 
as the author. (2) If he wrote it, it is hardly conceivable that he 
would have passed over in silence the discovery of the analytical 
method, the invention of Plato to which he attached so much 
importance*. 

There is nothing improbable in the conjecture that Proclus quoted 
the summary from a compendium of Eudemus' history made by some 
later writer: but as yet the question has not been definitely settled. 
All that is certain is that the early part of the summary must have 
been made up from scattered notices found in the great work of 
Eudemus. > [ 

Proclus refers to another work of Eudemus besides the history, 
viz. a book on T/i£ Angle {fiiffktoi/ Trept y^vias)\ Tannery assumes 
that this must have been part of the history, and uses this assumption 
to confirm his idea that the history was arranged according to subjects, 
not according to chronological order*. The phraseology of Proclus 
however unmistakably suggests a separate work; and that the 
history was chrmmlogicaily arranged seems to be clearly indicated by 
the remark of Simplicius that Eudemus **also counted Hippocrates 
among the more ancient writers " (cf toX^ TraXatoripo^^)*, 

The passage of Simplicius about the lunes of Hippocrates throws 
considerable light on the style of Eudemus' history. Eudemus wrote 
in a memorandum -like or summary manner (top v^o^vrifiaTLKov rpGirov 
rov l&ifB^fiovy when reproducing what he found in the ancient writers ; 
sometimes it is clear that he left out altogether proofs or constructions j 
of things by no means easy". 

Geminus. 

The discussions about the date and birthplace of Geminus form a 
whole literature, for an account of which I must refer the reader to the (];• 
recent edition by Manitius of Gemini elementa astronamiae (Teubner, 
1898)'. It must suffice here to state the general conclusion arrived at 
by Manitius^^ Though the name looks'^like a Latin name (Geminus), 

^ Diog. Laertins, IX. 40, p. 337, ed. Cobet * Proclus, p. 64, 8. 

' Proclus, p. an, 19 sqq. ; the passage is quoted above, p. 36. 

^ ibid. p. 135, 8. ' Tannery, La Ghmitrie grtcque^ p. 36. 

* Simplicius, ed. Diels, p. 69, 33. ' ibid, p. 60, 39. 

* Cf. Simplicius, p. 63, 19 sqq. ; p. 64, 35 saq. ; also Usener's note " de supplendis 
Hippocratis quas omisit Eudemus constructionibus *' added to Diels' pre£u:e, pp. xziu — xxvL 

* See the appendix to this edition, pp. 337 — 353. ^* pp. 351, 353. 



CH. ivj PROCLUS AND HIS SOURCES 39 

the consistent appearance of it in Greek with the properispomenon 
accent (Fe/i&ov) leaves no room for doubt as to its Greek origin, 
especially as it is found in inscriptions with the spelling Fc^ielvo^. 
The name may be formed from the stem ycfi like ^Epylvo^ from ipy, 
*AX€f*i/o9 from oXef. Cf also the unmistakably Greek names 'IktIpo^, 
Koarufo^, Geminus, a Stoic philosopher, born probably in the island 
of Rhodes, was the author of a comprehensive work on the classifi- 
cation of mathematics, and also wrote, about 73-67 B.C., a not less 
comprehensive commentary on the meteorological textbook of his 
teacher Posidonius of Rhodes. 

It is the former work in which we are specially interested here. 
Though Proclus made great use of it, he does not mention its title, 
unless we may suppose that, in the passage (p. 177, 24) where, after 
quoting from Geminus a classification of lines which never meet, he 
says, •* these remarks I have selected from the il>iXo/ea\la of Geminus," 
^iXoxaXia is a title or an alternative title. Pappus however quotes a 
work of Geminus *'on the classification of the mathematics" (iv r^ 
w€pi T^9 r£v fiaOrffjidrmv Taf6a)9)^ while Eutocius quotes from "the 
sixth book of the doctrine of the mathematics " {iv r^ iKnp t§9 r&v 
IJLaOfifiartov OempUn;)*. Tannery* pointed out that the former title 
corresponds well enough to the long extract* which Proclus gives in 
his first prologue, and also to the fragments contained in the Anonymi 
variae colUctiones published by Hultsch at the end of his edition of 
Heron* ; but it does not suit most of the other passages borrowed by 
Proclus. The correct title was therefore probably that given by 
Eutocius, The Doctrine, or Theory, of the Mathematics ; and Pappus 
probably refers to one particular portion of the work, say the first 
Book. If the sixth Book treated of conies, as we may conclude from 
Eutocius, there must have been more Books to follow, because Proclus 
has preserved us details about higher curves, which must have come 
later. If again Geminus finished his work and wrote with the same 
fulness about the other branches of mathematics as he did about 
geometry, there must ha(re been a considerable number of Books 
altogether. At all events it seems to have been designed to give 
a complete view of the whole science of mathematics, and in fact to 
be a sort of encyclopaedia of the subject. 

I shall now indicate first the certain, and secondly the probable, 
obligations of Proclus to Geminus, in which task I have only to follow 
van Pesch, who has embodied the results of Tittel's similar inquiry also*. 
I shall only omit the passages as regards which a case for attributing 
them to Geminus does not seem to me to have been made out. 

First come the following passages which must be attributed to 
Geminus, because Proclus mentions his name: 

(i) (In the first prologue of Proclus^) on the division of mathe- 

• Pappus, ed. Hultsch, p. 1026, 9. • Apollonius, ed. Heiberg, vol. ii. p. 170. 
' Tannery, La GionUtrU grecque, pp. i8, 19. * Proclus, pp. 38, i — 41, 8. 

• Heron, ed. Hultsch, pp. 246, 16--249, '^• 

• Van Pesch, De Procti fontibus, pp. 97—113. The disserUtion of Tittel is entitled De 
Gemini Stoici studiis nuUhimaticU ( 1 895). 

7 Proclus, pp. 38, I — 43, 8, except the allusion in p. 41, 8 — 10, to Ctesibius and Heron and 



40 INTRODUCTION [ch. iv 

matical sciences into arithmctiCt geometry, mechanics, astronoiny, 
optics, geodesy, canonic (science of musical harmony^ and logistic 
(apparently arithmetical problems); 

(2) (in the note on the definition of a straight line) on the 
classification of lines (including curves) as simple (straight or circular) 
and mixed, composite and incomposite, uniform (o^oiofttp*!^) and 
non-uniform {avo^oiop€p€U\ lines "about solids" and lines produced 
by cutting solids, including conic and spiric sections' ; 

(3) (i^ the note on the definition of a plane surface) on similar 
distinctions extended to surfaces and solids* ; 

(4) (in the note on the definition of parallels) on lines which 
do not meet {acvpirrmTOi) but which are not on that account 
parallel, e,g a curve and its asymptote, showing that the property of 
not nueling does not make lines parallel — a favourite observation of 
Geminus — and^ incidentally, on bounded lines or those which mciose a 
figure and those which do not* ; 

(5) (in the same note) the definition of parallels given by 
Posidonius*; 

(6) on the distinction between postulates and axioms, the futility 
of trying to prove axioms, .as ApoUonius tried to prove Axiom i, and 
the equal incorrectness of assuming what really requires proof, "as 
Euclid did in the fourth postulate [equality of right angles] and in 
the fifth postulate [the parallel- postulate]^" ; 

(7) on Postulates i, 2, 3, which Geminus makes depend on the 
idea of a straight line being described by the motion of a point* ; 

(8) (in the note on Postulate 5) on the inadmissibility in geometry 
of an argument which is merely plausible, and the danger in this 
particular case owing to the existence of lines which do converge 
ad infiniium and yet never meet^ ; 

(9) (in the note on L i) on the subject-matter of geometry, 
theorems, problems and Stopt^fioi (conditions of possibility) for 
problems* ; 

(10) (in the note on I. 5) on a generalisation of I. 5 by Geminus 
through the substitution for the rectilineal base of ** one uniform line 
(curve)/' by means of which he proved that the only "uniform lines" 

their pneumatic devices {BavnarowoiXiHi), as regards which Proclus' authority may be Pappus 
(yni. p. 1094, 74 — ay) who uses very similar exjpressioiis. Heron, even if not later tnan 
ueminus, could hardly have been included in a historical work bv him. Perhaps Geminus 
may have referred to Ctesibius only, and Proclus may have inserted "and Heron*' himself. 

* Prochis, pp. 103, «i— 107, 10; pp. Ill, I— 113, 3. 

* tM, pp. riy, 14—110, IS, where perhaps in the passage pp. 117, 33 — ii8, 93 we may 
have Gemmus' own words. 

* iMd. pp. 176, 18 — 177, 95; perhaps also p. 175. The note ends with the words 
"These things too we have selected from Geminus* ^iXoiraXki for the elucidation of the 
matters in question.** Tannery ^p. 17) takes these words coming at the end of the commen- 
tary on the definitions as referring to the whole of the portion of the commentary dealing 
with the definitions. Van Pesch properly regards them as only applying to the note on 
paraiUis, This seems to me clear from the use of the word too (ro^'aOra ir aQ. 

* Proclus, p. 176, 5—17. 

* ihid, pp. 178— 18s, 4; pp. 183, 14— 184, 10; cf. p. 188, 3— II. 

* ibid. p. 185, 6 — 15. 

' ibid. p. 193, 5 — 19. * iHd, pp. loo, ii — aoi, 35. 



CH. iv] PROCLUS AND HIS SOURCES 41 

» 

, [ (alike in all their parts) are a straight line, a circle, and a cylindrical 
t; helix^ 

(11) (in the note on I. lo) on the question whether a line is made 
up of indivisible parts (dfiepr}), as affecting the problem of bisecting 
a given straight line* ; 

(12) (in the note on I. 35) on topical, or /<?rKJ-theorems*, where 
the illustration of the equal parallelograms described between a 
hyperbola and its asymptotes may also be due to Geminus*. 

Other passages which may fairly be attributed to Geminus, though 
his name is not mentioned, are the following : 

(i) in the prologue, where there is the same allusion as in the 
passage (8) above to a remark of Aristotle that it is equally absurd to 
expect scientific proofs from a rhetorician and to accept mere plausi- 
bilities from a geometer* ; 

(2) a passage in the prologue about the subject-matter, methods, 
and bases of geometry, the latter including axioms and postulates* ; 

(3) another on the definition and nature of eletnents^ ; 

(4) a remark on the Stoic use of the term axiom for every simple 
statement {air6<f>avai^ awXriY ; 

(5) another discussion on theorems and problems*, in the middle 
of which however there are some sentences by Proclus himself". 

(6) another passage, in connexion with Def. 3, on lines including 
or not including a figure (with which cf part of the passage (4) 
above)" ; 

(7) a classification of different sorts of angles according as they 
are contained by simple or mixed lines (or curves)"; 

(8) a similar classification of figures", and of plane figures" ; 

(9) Posidonius' definition of a figure^* ; 

(10) a classification of triangles into seven kinds" ; 

(11) a note distinguishing lines (or curves) producible indefinitely 
or not so producible, whether forming a figure or not forming a 
figure (like the "single-turn spiral")^'; 

(12) passages distinguishing different sorts of problems", different 
sorts of theorems", and two sorts of converses (complete and partial)*; 

(13) the definition of the term "porism" as used in the title of 
Euclid's Porisms, as distinct from the other meaning of "corollary ""; 

(14) a note on the Epicurean objection to I. 20 as being obvious 
even to an ass" ; 

(15) a passage on the properties of parallels, with allusions to 

• Proclus, p. 251, 2— II. • ibid. pp. 277, 25 — 370, 11. 
» ibid, pp. 394, 11—395, a and p. 395, 13—21. * ibid, p. 395, 8—12. 

• ibid. pp. 33, 21—34. I. • ibid. pp. 57, 9—58, 3. 
' *^' PP- 72. 3— 75» 4- * if^' p. 77. 3—6- 

• ibid. pp. 77, 7—78, 13, and 79, 3—81, 4. " H>id. pp. 78, 13—79* «• 
^» ibid. pp. 102, 31—103, '8. " ibid. pp. 116, 7—117, iC 
" ibid, pp. 159, II — 160, 9. " ibid, pp. 161, 37—164, 6. 
" ibid. p. 143, 5— II. " ibid. p. 168, 4—13. 
*' ibid. p. 187, 19—37. w 1^^. pp. 220, 7 — 333, 14; also p. 330, 6—9. 
" ibid, pp. 344, 14—346, 13. « ibid, pp. 353, 5—354, 30. 
» ibid. pp. 301, 31—303, 13. « ibid, pp. 333, 4— 3«3. 3- 



4f INTRODUCTION [cw. iv 

Apollonms' C*?*kj, and the curves invented by Nicomedes, Hippias 
and Perseus' ; 

(t6) a passage on the parallel-postulate regarded as the converse 
of L ij\ 

Of the authors to whom Proclus was indebted in a less degree the 
most important is Apollonius of Perga. Two passages allude to his 
Ccni€s\ one to a work on irrationals*, and two to a treatise On /A# 
cocklias (apparently the cylindrical helix) by Apollonius*. But more 
important for our purpose are six references to Apollonius in connexion 
with elementary geometry, 

(i) He appears as the author of an attempt to explain the idea 
of a line (possessing length but no breadth) by reference to daily 
experience, e,g. when we tell someone to measure^ merely, the length 
of a road or of a wall*; and doubtless the similar passage showing 
how we may in like manner get a notion of a surface (without depth) 
is his also*. 

(2) He gave a new general definition of an angle*, . 

(3) He tried to prove certain axioms', and Proclus gives his I 
attempt to prove Axiom i, word for word"- J 

Proclus further quotes ; \ 

(4) Apollonius' solution of the problem in Eucl. L lo^ avoiding 
Euclid*s use of L 9", 

(5) his solution of the problem in l. 11, differing only slightly 
from Euclid's*', and 

(6) his solution of the problem in l. 23^. 
Heiberg" conjectures that Apollonius departed from Euclid's 

method in these propositions because he objected to solving problems 
of a more general, by means of problems of a more particular* 
character. Proclus however considers all three solutions inferior to 
Euclid's ; and his remarks on Apollonius' handling of these ele- 
mentary matters generally suggest that he was nettled by criticisms 
of Euclid in the work containing the things which he quotes from 
Apollonius, just as we conclude that Pappus was offended by the 
remarks of Apollonius about Euclid's incomplete treatment of the 
*• three- and four-line locus"." If this was the case, Proclus can hardly 
have got his information about these things at second-hand; and 
there seems to be no reason to doubt that he had the actual work of 
Apollonius before him. This work may have been the treatise 
mentioned by Marinus in the words ''Apollonius in his general 
treatise" (*A7roXXioi/io^ iv tJ icaOoKov irpayfjuireiay^ If the notice 
in the Fthrist^'' stating, on the authority of Thabit b. Qurra, that 

» Proclus, pp. 355, 10--356, 16. « ibid. p. 364. 9—12; pp. 3^4. ao— 365,4. 

» ibid. p. 71, lOjL-i*. 356, 8, 6. « ibid. p. 74, 23, 24. 

• ibid. pp. i&j, 5, 6, 14, 15. • ibid. p. 100, 5—19. 
' ibid. p. 114, ao— 15. • ibid. p. 123, 15—19 (cf. p. 114, 17, p. 115, 17). 

• ibid. p. 183. 13, 14. w ibid, pp. 194. as— '95f 5« 
" ibid. pp. 179, 16—180, 4. '• ibid, p. 281, 8—19. 
" ibid. pp. 335, 16—336, 5. " PkihUgus, vol. XLliI. p. 489. 
^ See above, pp. a, 3. >* Marinus in EucHdis Daia^ ed. Menge, p. S34, 16. 
» Fikrist^ tr. Sutcr, p. 19. 



i 



CH. IV] PROCLUS AND HIS SOURCES 43 

Apollonius wrote a tract on the parallel-postulate be correct, it may 
have been included in the same work. We may conclude generally 
that, in it, Apollonius tried to remodel the beginnings of geometry, 
reducing the number of axioms, appealing, in his definitions of lines, 
surfaces etc., more to experience than to abstract reason, and 
substituting for certain proofs others of a more general character. 

The probabilities are that, in quoting from the tract of Ptolemy in 
which he tried to prove the parallel-postulate, Proclus had the actual 
work before him. For, after an allusion to it as **a certain book**' 
he gives two long extracts*, and at the beginning of the second 
indicates the title of the tract, '* in the (book) about the meeting of 
straight lines produced from (angles) less than two right angles," as 
he has very rarely done in other cases. 

Certain things from Posidonius are evidently quoted at second- 
hand, the authority being Geminus (e.g. the definitions of figtire and 
parallels) ; but besides these we have quotations from a separate work 
which he wrote to controvert Zeno of Sidon, an Epicurean who had 
sought to destroy the whole of geometry*. We are told that Zeno 
had argued that, even if we admit the fundamental principles (apxai) 
of geometry, the deductions from them cannot be proved without the 
admission of something else as well, which has not been included in 
the said principles*. On I. i Proclus gives at some length the argu- 
ments of Zeno and the reply of Posidonius as regards this proposition*. 
In this case Zeno's "something else" which he considers to be 
assumed is the fact that two straight lines cannot have a common 
segment, and then, as regards the "proof" of it by means of the 
bisection of a circle by its diameter, he objects that it has been 
assumed that two circumferences (arcs) of circles cannot have a 
common part Lastly, he makes up, for the purpose of attacking it, 
another supposed " proof" of the fact that two straight lines cannot 
have a common part. Proclus appears, more than once, to be quoting 
the actual words of Zeno and Posidonius ; in particular, two expres- 
sions used by Posidonius about "the acrid Epicurean" {jov Bpifiifv 
*EinKovp€iopy and his "misrepresentations" (Uoa-eiBwpti^ <fnfai rov 
Zi^vtDva avKo<f>avTelvy, It is not necessary to suppose that Proclus 
had the original work of Zeno before him, because Zeno's arguments 
may easily have been got from Posidonius' reply ; but he would 
appear to have quoted direct from the latter at all events. 

The work of Carpus mechanicus (a treatise on astronomy) quoted 
from by Proclus* must have been accessible to him at first-hand, 
because a portion of the extract from it about the relation of theorems 
and problems* is reproduced word for word. Moreover, if he were not 
using the book itself, Proclus would hardly be in a position to question 
whether the introduction of the subject of theorems and problems 

• Proclus, p. 191, 13. * ibid, pp. 361, 14— 3^3» ^8» PP- l^hy 7— 3^7» «7- 

• ibid, p. 100, 1 — 3. * ibid, pp. 199, 11 — 100, i. 

• ibid, pp. 314, 18 — 315, 13; pp. 116, 10 — 118, II. 

• ibid, p. 216, 21. ' ibid, p. 218, i. 

• ibid, pp. 341, 19 — 343, II. * ibid, pp. 143, 33—343, 11. 



L 



44 INTRODUCTION [cH. iv 

was opportune in the place where it was found (et fiiv mari tcaipiv f} 

It is of course evident that Proclus had before him the original 
works of PlatOj Aristotle. Archimedes and Plotinus, as well as the 
Xvfifittcrd of Porphyry and the works of his master Syrianus (o i5^cTf/>o? 
ica0iiyffimpy, from whom he quotes in his note on the definition of an 
angle. Tannery also points out that he must have had before him a 
group of works representing the Pythagorean tradition on its mystic, 
as distinct from its mathematical, side, from Phitolaus downwards, and 
comprising the more or less apocryphal Upoi; \oyos of Pythagoras, the 
Oracles (Xojta), and Orphic verses". 

Besides quotations from writers whom wc can identify with more 
or less certainty, there are many other passages which are doubtless 
quoted from other commentators whose names we do not know. A 
list of such passages is given by van Pesch*, and there is no need to 
cite them here. 

Van Pesch also gives at the end of his work ' a convenient list of 
the books which, as the result of his investigation, he deems to have 
been accessible to and directly used by Proclus. The list is worth 
giving here, on the same ground of convenience. It is as follows: ^ 

Eudemus : history of giomttry. 

Gem in us : the theory of the mathematical sciejtces, 
r Heron : commmlary on the Elements of Euclid. 

Porphyry: „ „ „ . 

Pappus: „ „ „ j 

Apollonius of Perga : a work relating to elementary geometty* 

Ptolemy: on the parallel-postulate. 

Posidonius : a book controverting Zeno of Sidon. 

Carpus ; astronomy, 

Syrianus : a discussion on the angle. 

Pythagorean philosophical tradition, 

Plato s works » 

Aristotle's works. 

Archimedes' works. 

Plotinus: Enneades, 
Lastly we come to the question what passages, if any, in the 
commentary of Proclus represent his own contributions to the subject 
As we have seen, the onus probandi must be held to rest upon him 
who shall maintain that a particular note is original on the part of 
Proclus. Hence it is not enough that it should be impossible to point 
to another writer as the probable source of a note ; we must have a 
positive reason for attributing it to Proclus. The criterion must there- 
fore be found either (i) in the general terms in which Proclus points 
out the deficiencies in previous commentaries and indicates the 
respects in which his own will differ from them, or (2) in specific 
expressions used by him in introducing particular notes which may 

^ Proclus, p. 341, 21, 33. ' ibid, p. 133, 19. 

' Tannery, La Giomitrii gncque^ pp. 35, 36. 

* Van Pesch, Di Proclifontibus^ p. 139. • ibid. p. 155. 



u 



CH. iv] PROCLUS AND HIS SOURCES 45 

indicate that he is giving his own views. Besides indicating that he 
paid more attention than his predecessors to questions requiring 
deeper study (ro irpayfunei^he^) and " pursued clear distinctions " 
(to evBiaiperop fji^rahiw/covra^y — by which he appears to imply that 
his predecessors had confused the different departments of their 
commentaries, viz. lemmas, cases, and objections {iparaaci^y — Proclus 
complains that the earlier commentators had failed to indicate the 
ultimate grounds or causes of propositions*. Although it is from 
Geminus that he borrowed a passage maintaining that it is one of the 
proper functions of geometry to inquire into causes (r^v airiav icaX 
TO tik riy, yet it is not likely that Geminus dealt with Euclid's 
propositions one by one ; and consequently, when we find Proclus, on 
\\ I. 8, 16, 17, 18, 32, and 47', endeavouring to explain causes, we have 
good reiEison to suppose that the explanations are his own. 

Again, his remarks on certain things which he quotes from Pappus 
can scarcely be due to anyone else, since Pappus is the latest of the 
commentators whose works he appears to have used. Under this 
head come 

(i) his objections to certain new axioms introduced by Pappus*, 

(2) his conjecture as to how Pappus came to think of his alterna- 
tive proof of I. 5^ 

(3) an addition to Pappus' remarks about the curvilineal angle 
which is equal to a right angle without being one*. 

The defence of Geminus against Carpus, who combated his view 
of theorems and problems, is also probably due to Proclus*, as well as 
an observation on I. 38 to the effect that I. 35 — 38 are really compre- 
hended in VI. I as particular cases^*. 

Lastly, we can have no hesitation in attributing to Proclus himself 
(i) the criticism of Ptolemy's attempt to prove the parallel-postulate >S 
and (2) the other attempted proof given in the same note^ (on I. 29) 
and assuming as an axiom that '' if from one point two straight lines 
forming an angle be produced ad infinitum the distance between them 
when so produced ad infinitum exceeds any finite magnitude (i.e. 
length)," an assumption which purports to be the equivalent of a 
statement in Aristotle". It is introduced by words in which the 
writer appears to claim originality for his proof: "To him who 
desires to see this proved {KaTaa'tc€va^6fi€vop) let it be said by us 
(Keyiado) irap ^fjLoSv)" etc." Moreover, Philoponus, in a note on 
Aristotle's Anal, post, I. 10, says that **the geometer (Euclid) assumes 
this as an axiom, but it wants a great deal of proof, insomuch that 
both Ptolemy and Proclus wrote a whole book upon it"." 

> Proclus, p. 84, 13, p. 432, 14, 
• ibid, - -- -- 



CHAPTER V. 



THE TEXT", 

It is well known that the title of Simson's edition of Euclid (first 
brought out in Latin and English in 1756) claims that, in it, "the 
errors by which Theon, or others, have long ago vitiated these books 
are corrected, and some of Euclid's demonstrations arc restored " ; and 
readers of Simson's notes are familiar with the phrase^ used, where 
anything in the text does not seem to htm satisfactory, to the effect 
that the demonstration has been spoiled, or things have been interpo- 
lated or omitted, by Theon "or some other unskilful editor/' Now 
most of the MSS. of the Greek text prove by their titles that they 
proceed from the recension of the Elements by Theon ; they purport 
to be either " from the edition of Theon " [ix t^9 ^iti^vo^ €xSo<r€<a?) or 
** from the lectures of Theon *' (agro avvovaimu rov %itapo^). This was 
Theon of Alexandria (4th c. A.D,) who also wrote a commentary on 
Ptolemy, in which there occurs a passage of the greatest importance 
in this connexion*: "But that sectors in equal circles are to one 
another as the angles on which they stand Ans been proved by me in 
my edition of the Elements at tfu end of the sixth book!' Thus Theon 
himself says that he edited the Elements and also that the second part 
of VI. 33, found in nearly all the MSS,, is his addition. 

This passage is the key to the whole question of Theon*s changes 
in the text of Euclid ; for, when Peyrard found in the Vatican the 
MS. 190 which contained neither the words from the titles of the other 
MSS. quoted above nor the interpolated second part of VI. 33, he was 
justified in concluding, as he did, that in the Vatican MS, we have an 
edition more ancient than Theon 's. It is also clear that the copyist 
of P,or rather of its archetype, had before him the two recensions and 
systematically gave the preference to the earlier one ; for at XIII. 6 in 
P the first hand has added a note in the margin 1 ** This theorem is 
not given in most copies of the new editimt, but is found in those of 
the old," Thus we are more fortunate than Simson, since our 
judgment of Theon*s recension can be formed on the basis, not of 
mere conjecture, but of the documentary evidence afforded by a 
comparison of the Vatican MS. just mentioned with what we may 
conveniently call, after Heiberg, the Theonine MSS* 

* The materkl for the whole af this chapter is Uken firom Heiberg's cdtticm of the 
EUmenii^ introduction to vol. V., And itotn tbe same schoUr's Uiierargtsatuhiiuk^ SittdUn 
iiber Euklidy p. 174 sqq. ^d Fi^ralipomtna fv Euklidtn Htrma^ XXXV I It.^ 1905. 

* I. p. aoj ed. Hainia = p. 50 etL }^iSA\. 



CH. v] THE TEXT 47 

The MSS. used for Heibei^s edition of the Elements are the 
following : 

(i) P = Vatican MS. numbered 190, 4to, in two volumes (doubt- 
less one originally) ; loth c. 

This is the MS. which Peyrard was able to use ; it was sent from 
Rome to Paris for his use and bears the stamp of the Paris Imperial 
Library on the last page. It is well and carefully written. There are 
corrections some of which are by the original hand, but generally in 
paler ink, others, still pretty old, by several different hands, or by one 
hand with different ink in different places (P m. 2), and others again 
by the latest hand (P m. rec). It contains, first, the Elements I. — XIII. 
with scholia, then Marinus' commentary on the Data (without the 
name of the author), followed by the Data itself and scholia, then the 
Elements XIV., XV. (so called), and lastly three books and a part of a 
fourth of a commentary by Theon €t9 toi)? irpox^ipov^ icavova^ OtoXc- 
liaiov. 

The other MSS. are " Theonine." 

(2) F = MS. XXVIII, 3, in the Laurentian Library at Florence, 4to ; 

loth c. 
This MS. is written in a beautiful and scholarly hand and contains 
the Elements I. — XV., the Optics and the Phaenomena, but is not well 
preserved. Not only is the original writing renewed in many places, 
where it had become faint, by a later hand of the i6th c, but the same 
hand has filled certain smaller lacunae by gumming on to torn 
pages new pieces of parchment, and has replaced bodily certain 
portions of the MS., which had doubtless become iil^ible, by fresh 
leaves. The larger gaps so made good extend from Eucl. vii. 12 to 
IX. 1 5, and from XII. 3 to the end ; so that, besides the conclusion of the 
Elements, the Optics and Phaenomena are also in the later hand, and we 
cannot even tell what in addition to the Elements I. — XIII. the original 
MS. contained. Heiberg denotes the later hand by ^ and observes 
that, while in restoring words which had become faint and filling up 
minor lacunae the writer used no other MS., yet in the two larger 
restorations he used the Laurentian MS. xxviil, 6, belonging to the 
13th — 14th c. The latter MS. (which Heiberg denotes by f) was 
copied from the Viennese MS. (V) to be described below. 

(3) B = Bodleian MS., D'Orville X. i inf. 2, 30, 4to ; A.D. 888. 
This MS. contains the Elements I. — xv. with many scholia. Leaves 

15 — 118 contain L 14 (from about the middle of the proposition) to 
the end of Book VI., and leaves 123 — 387 (wrongly numbered 397) 
Books VII. — XV. in one and the same elegant hand (9th c). The 
leaves preceding leaf 1 5 seem to have been lost at some time, leaves 
' 6 to 14 (containing Elem, I. to the place in I. 14 above referred to) 
being carelessly written by a later hand on thick and common parch- 
ment (13th c). On leaves 2 to 4 and 122 are certain notes in the 
hand of Arethas, who also wrote a two-line epigram on leaf 5, the 
greater part of the scholia in uncial letters, a few notes and corrections, 
and two sentences on the last leaf, the first of which states that the 
MS. was written by one Stephen clericus in the year of the world 6397 



{ 



48 INTRODUCTION [cm. v 

(^ 888 A.D,)j while the second records Arethas* own acqiiisition of it 
Arethas lived from, say, 865 to 939 A.D. He was Archbishop of 
Caesarea and wrote a commentary on the Apocalypse, The portions 
of his library which survive are of the greatest interest to palaeography 
on account of his exact notes of dates, names of copyists, prices of 
parchment etc. It is to him also that we owe the famous Plato MS. 
from Pat m OS (Cod. Clarkianus) which was written for him in November 
895 », 

(4) V = Viennese MS. Philos, Gn No. 103 ; probably 12th c 
This MS* contains 292 leaves. Eucl Ekments h — XV. occupying 
leaves 1 to 254, after which come the Optics (to leaf 271), the 
PAacn(^fn€fta {mxyWlTLX^d at the end) from leaf 272 to leaf 2S2, and lastly 
scholia^ on leaves 283 to 292^ also imperfect at the end. The different 
materia] used for different parts and the varieties of handwriting make 
it necessary for Heiberg to discuss this MS, at some length*. The 
handwriting on leaves 1 to 183 (Book h to the middle of X. 105) and 
on leaves 203 to 234 (from XL 31, towards the end of the proposition, 
to XIII. 7, a few lines down) is the same ; between leaves 184 and 202 
there arc two varieties of handwriting, that of leaves 184 to '189 and 
that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in 
the same handwriting, changes first gradually into that of leaves 184 
to 1 89 and then (verso) into a third more rapid cursive writing which 
is the same as that of the greater part of the scholia, and also as that 
of leaves 243 and 282, althoughj as these leaves are of different 
material, the look of the writing and of the ink seems altered 
There are corrections both by the first and a second hand, and scholia 
by many hands. On the whole, in spite of the apparent diversity of 
handwriting in the MS., it is probable that the whole of it was written 
at about the same time, and it may (allowing for changes of material, 
ink etc) even have been written by the same man. It is at least 
certain that, when the Laurentian M.s* xxviii^O was copied from it, the 
whole MS. was in the condition in which it is now, except as regards 
the later scholia and leaves 283 to 292 which are not in the laurentian 
MS.j that MS. coming to an end where the F/taetwrnena breaks off 
abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c. 
But it was apparently in two volumes originally, the first-con- 
sisting of leaves i to 183 ; and it is certain that it was not all copied 
at the same time or from one and the same original. For leaves 
1 84 to 202 were evidently copied from two MSS. different both from 
one another and from that from which the rest was copied. Leaves 
1 84 to the middle of leaf 1 89 (recto) must have been copied from a 
MS, similar to P^ as is proved by similarity of readings, though not 
from P itself The rest, up to leaf 202, were copied from the Bolc^na 
MS, (b) to be mentioned below. It seems clear that the content of 
leaves 184 to 202 was supplied from other MSS. because there was a 
lacuna in the original from which the rest of V was copied ] 



] 



< 



\ 



^ See Pauly-WUsowa, Rial-EncydopadU der class. AlUrtumtunssensckaftiHt voL IL, 18961 
eibeig, vol v. pp. zxix— xxxiii. 



p. 67^^ 



r' 



cav] THE TEXT 49 

Heiberg sums up his conclusions thus. The copyist of V first 
copied leaves i to 183 from an original in which two quaterniones 
were missing (covering from the middle of Eucl. x. 105 to near the 
end of XL 31). Noticing the lacuna he put aside one quatemio of the 
parchment used up to that point Then he copied onwards from 
the end of the lacuna in the original to the end of the Phamomena, 
After this he looked about him for another MS. from which to fill up 
the lacuna ; finding one, he copied from it as far as the middle of leaf 
189 (recto). Then, noticing that the MS. from which he was copying 
was of a different class, he had recourse to yet another MS. from which 
he copied up to leaf 202. At the sanre time, finding that the lacuna 
was longer than he had reckoned for, he had to use twelve more 
leaves of a different parchment in addition to the quatemio which he 
had put aside. The whole MS. at first formed two volumes (the first 
containing leaves i to 183 and the second leaves 184 to 282); then, 
after the last leaf had perished, the two volumes were made into one 
to which two more quaterniones were also added. A few leaves of the 
latter of these two have since perished. 

(5) b s MS. numbered 18 — 19 in tlie Communal Library at 

Bologna, in two volumes, 4to; nth c. 
This MS. has scholia in the margin written both by the first hand 
and by two or three later hands ; some are written by the latest hand, 
Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 
14th c) who owned the MS. at one time. It contains {a) in 14 quater- 
niones the definitions and the enunciations (without proofs) of the 
Elements I. — XIIL and of the Data^ {b) in the remainder of the 
volumes the Proem to Geometry (published among the Variae 
Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) 
followed by the Elements I.— XIII. (part of XIII. 18 to the end being 
missing), and then by part of the Data (from the last three words of 
the enunciation of Prop. 38 to the end of the penultimate clause in 
Prop. 87, ed. Menge). From XI. 36 inclusive to the end of xil. this 
MS. appears to represent an entirely different recension. Heiberg is 
compelled to give this portion of b separately in an appendix. He 
conjectures that it is due to a Byzantine mathematician who thought 
j- Euclid's proofs too long and tiresome and consequently contented 
11 himself with indicating &e course followed^ At the same time this 
Byzantine must have had an excellent MS. before him, probably of the 
. ante-Theonine variety of which the Vatican MS. 190 (P) is the sole 
representative. 

(6) p = Paris MS. 2466, 4to; 12th c. 

This manuscript is written in two hands, the finer hand occupying 
leaves i to 53 (recto), and a more careless hand leaves 53 (verso) to 
64, which are of the same parchment as the earlier leaves, and leaves 
I 65 to 239, which are of a thinner and rougher parchment showing 
traces of writing of the 8th — 9th c. (a Greek version of the Old 
Testament). The MS. contains the Elements I. — xill. and some scholia 
after Books xi., XII. and Xlll. 

> Ztiischrifi fiir Math, u. Physik^ XXIX., hist.-litt. Abtheilung, p. 13. 
H. E. 4 



I- 



50 INTRODUCTION [cK, v 

(7) q = Pans MS. 2344, folio ; 12th c 
It is written by one hand but includes scholia by many hands. 

On leaves l to 16 (recto) are scholia with the same title as that found 
by Wachsmuth in a Vatican MS. and relied upon by him to prove that 
Proclus continued his commentaries beyond Book L' Leaves 17 to 
357 contain the Elements L — XIIL (except that there is a lacuna from 
the middle of VIIL 25 to the Sxffetrt*^ of IX, 14); before Books VIL and 
X, there are some leaves filled with scholia only* and leaves 358 to 366 
contain nothing but scholia, 

(8) Heiberg also used a palimpsest in the British Museum (Add, 
1 72 11). Five pages are of the 7th — 8th c, and are contained (leaves 
49 — S3) in the second volume of the Syrian ms. Brit. Mus. 687 of the 
9th c, ; half of leaf 50 has perished » The leaves contain various frag- 
ments from Book X. enumerated by Heiberg, Vol. III., p, v, and nearly 
the whole of X II I, 14, 

Since his edition of the Elements was published, Heiberg has 
collected further material bearing on the history of the text*. Besides 
giving the results of further or new examination of MSS,. he has 
collected the fresh evidence contained in an-Nairizl's comment a r>\ 
and particularly in the quotations from Heron *3 commentary given in 
it (often word for word^ which enable us in several cases to trace I 
differences between our text and the text as Heron had it, and to 
identify some interpolations which actually found their way into the 
text from Heron's commentary itself; and lastly he has dealt with 
some valuable fragments of ancient papyri which have recently come 
to light, and which are especially important in that the evidence drawn 
from them necessitates some modification in the views expressed in 
the preface to VoL V, as to the nature of the changes made in Theon's 
recension, and in the principles laid down for differentiating between 
Theon's recension and the original text, on the basis of a comparison 
between P and the Theonine MSS. alone* 

The fragments of ancient papyri referred to are the following, 

I* Pafyrus Herculanensis No. 1061* 

This fragment quotes Def. i j of Book l. in Greek, and omits the 
words ^ ^aX^trat 7r€pi<f>ip€ta^ "which is called the circumference/' 
found in all our MSS., and the further addition ^/>ot rtju roi) /WKXav 
TTipitf^tp^tav also found in practically all the MSS. Thus Heibei^'s 
assumption that both expressions are interpolations is now confirmed 
by this oldest of all sources, 

2. T^ Oxyrhymhus Papyri 1. p. 58, No. XXIX* of the 3rd or 4th c. 

This fragment contains the enunciation of Eucl IL 5 (with figure, 
apparently without letters, immediately following, and not, as usual in 
our MSS., at the end of the proof) and before it the part of a word"" 
vepiexofie belonging to II. 4 (with room for -yf» opOo^vUp* iirep l^i 

^ [tit rjk T9O EAxXcldov rrocxcM wpdXa^ifiaifi/uwa 4k rQif np6ffXov aw^fdimif mi car* fri- 
n^^. Cf p. 3a, note 8, above. 

' Heibeig, Faratipomtna mm EukHd in Hermes^ xxxviiu, 15^3, pp. 46 — 74, 161 — aoi, 

' Described by Heibeig in Otfersigt aver dit kmgi. damJki VuUmkabemis Siiskttbs 
Forhandtingpr^ 1900, p. 161. 



. . 



(" 



CH.V] THE TEXT 51 

Set^ai and a stroke to mark the end), showing that the fragment had 
not the Porism which appears in all the Theonine MSS. and (in a later 
hand) in P, and thereby confirming Heiberg's assumption that the 
Porism was due to Theon. 

3. A fragment in Fayum towns and ttieir papyri^ p. 96, No. IX. of 
2nd or 3rd c 

This contains I. 39 and I. 41 following one another and almost 
complete, showing that I. 40 was wanting, whereas it is found in all 
the MSS. and is recognised by Proclus. Moreover the text of the 
beginning of I. 39 is better than ours, since it has no double hiopur/AS^ 
but omits the first (" I say that they are also in the same parallels ") 
and has " and** instead of **/or let ^Z> be joined " in the next sentence. 
It is clear that I. 40 was interpolated by someone who thought there 
ought to be a proposition following I. 39 and related to it as I. 38 is 
related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses I. 40, 
and therefore was not likely to include it The same interpolator 
failed to realise that the words "let AD he joined" were part of the 
i/cOeai^ or setting-out, and took them for the tcaraa-tcewj or " construc- 
tion " which generally follows the hiopurfio^ or " particular statement " 
of the conclusion to be proved, and consequently thought it necessary 
to insert a Siopurfio^ before the words. 

The conclusions drawn by Heiberg from a consideration of 
particular readings in this papyrus along with those of our MSS. will 
be referred to below. 

We now come to the principles which Heiberg followed, when 
preparing his edition, in differentiating the original text from the 
Theonine recension by means of a comparison of the readings of P 
and of the Theonine MSS. The rules which he gives are subject to a 
certain number of exceptions (mostly in cases where one Ms. or the 
other shows readings due to copyists' errors), but in general they may 
be relied upon to give conclusive results'. 

The possible alternatives which the comparison of P with the 
Theonine MSS. may give in particular passages are as follows : 

I. There may be agreement in three different degrees. 

(i) P and all the Theonine MSS. may agree. 

In this case the reading common to all, even if it is corrupt or 
interpolated, is more ancient than Theon, i.e. than the 4th c. 

(2) P may agree with some (only) of the Theonine MSS. 

In this case Heiberg considered that the latter give the true 
reading of Theon's recension, and the other Theonine MSS. have 
departed from it 

(3) P and one only of the Theonine MSS. may agree. 

In this case too Heiberg assumed that the one Theonine MS. which 
agrees with P gives the true Theonine reading, and that this rule even 
supplies a sort of measure of the quality and faithfulness of the 
Theonine MSS. Now none of them agrees alone with P in preserving 
the true reading so often as F. Hence F must be held to have pre- 
served Theon 's recension more faithfully than the other Theonine MSS.; 
and it would follow that in those portions where F fails us P must 

4—2 



L 



$2 INTRODUCTION [ch. 

cany rather more weight even though it may differ from the Theonine 
MSS. BVpq. (Heiberg gives many examples en proof of this, as of his 
main rules generally, for which reference must be made to his Prole- 
gomena in Vol V.) The specially close relation of F and P is also 
illustrated by passages in which they have the same errors \ the 
explanation of these common errors (where not due to accident) is 
found by Heiberg in the supposition that they existed, but were not 
noticed by Theon, in the original copy in which he made his changes. 

Although however F is by far the best of the Theonine MSS., there 
are a considerable number of passages where one of the others (B, V, 
p or q) alone with P gives the genuine reading of Theon's recension. 

As the result of the discovery of the papyrus fragment containing 
I. 39, 41, the principles above enunciated under (2) and (3) are found 
by Heiberg to require some qualification. For there is in some cases 
a remarkable agreement between the papyrus and the Theonine MSS. 
(some or all) as against R This shows that Theon took more trouble 
to follow older MSS,, and made fewer arbitrary changes of his own, 
than has hitherto been supposed. Next, when the papyrus agrees 
with some of the Theonine MSS. against P, it must now be held that 
these MSS. (and not, as formerly supposed, those which agree with P) 
give the true reading of Theon. If it were otherwise, the agreement 
between the papyrus and the Theonine MSS. would be accidental: but 
it happens too often for this. It is clear also that there must have 
been contamination between the two recensions ; otherwise, whence 
could the Theonine MSS, which agree with P and not with the papyrus 
have got their readings? The influence of the P class on the Theonine 
F is especially marked. 

11. There may be disagreement between P and all the Theonine 
MSS. 

The following possibilities arise. 

(1) The Theonine MSS. differ also among themselves* 

In this case Heiberg considered that P nearly always has the true 
reading, and the Theonine MSS, have suffered interpolation in different 
ways after Theon's time. 

(2) The Theonine Mss. all combine against P. 

In this case the explanation was assumed by Heiberg to be one or 
other of the following. 

(a) The common reading is due to an error which cannot be 
imputed to Theon (though it may have escaped him when putting 
together the archetype of his edition); such error may either have 
arisen accidentally in all alike, or (more frequently) may be 
referred to a common archetype of all the MSS. 

(/3) There may be an accidental error in P ; e.g. something 
has dropped out of P in a good many places, generally through 

(<y) There may be words interpolated in P. 
(0) Lastly, we may have in the Theonine MSS. a change made 
by Theon himself, 
(The discovery of the ancient papyrus showing readings agreeing 



i 



CH. v] THE TEXT 53 

with some, or with all, of the Theonine MSS. against P now makes it 
necessary to be very cautious in applying these criteria.) 

It is of course the last class (8) of changes which we have to 
investigate in order to get a proper idea of Theon's recension. 

Heiberg first observes, as regards these, that we shall find that 
Theon, in editing the Elements, altered hardly anything without some 
reason, often inadequate according to our ideas, but still some reason 
which seemed to him sufficient Hence, in cases of very slight differ- 
ences where both the Theonine MSS. and P have readings good and 
probable in themselves, Heiberg is not prepared to put the differences 
down to Theon. In those passages where we cannot see the least 
reason why Theon, if he had the reading of P before him, should have 
altered it, Heiberg would not at once assume the superiority of P 
unless there was such a consistency in the differences as would indicate 
that they were due not to accident but to design. In the absence of 
such indications, he thinks that the ordinary principles of criticism 
should be followed and that proper weight should be attached to the 
antiquity of the sources. And it cannot be denied that the sources of 
the Theonine version are the more ancient. For not only is the 
British Museum palimpsest (L), which is intimately connected with 
the rest of our MSS., at least two centuries older than P, but the other 
Theonine MSS. are so nearly allied that they must be held to have 
had a common archetype intermediate between them and the actual 
edition of Theon ; and, since they themselves are as old as, or older 
than P, their archetype must have been much older. Heiberg gives 
(pp. xlvi, xlvii) a list of passages where, for this reason, he has 
followed the Theonine MSS. in preference to P. 

It has been mentioned above that the copyist of P or rather of its 
archetype wished to give an ancient recension. Therefore (apart from 
clerical errors and interpolations) the first hand in P may be relied 
upon as giving a genuine reading even where a correction by the first 
hand has been made at the same time. But in many places the first 
hand has made corrections afterwards ; on these occasions he must 
have used new sources, e.g. when inserting the scholia to the first 
Book which P alone has, and in a number of passages he has made 
additions from Theonine MSS. 

We cannot make out any " family tree " for the different Theonine 
MSS. Although they all proceeded from a common archetype later 
than the edition of Theon itself, they cannot have been copied one 
from the other ; for, if they had been, how could it have come about 
that in one place or other each of them agrees alone with P in pre- 
serving the genuine reading } Moreover the great variety in their 
agreements and disagreements indicates that they have all diverged 
to about the same extent from their archetype. As we have seen that 
P contains corrections from the Theonine family, so they show correc- 
tions from P or other MSS. of the same family. Thus V has part of 
the lacuna in the MS. from which it was copied filled up from a MS. 
similar to P, and has corrections apparently derived from the same ; 
the copyist, however, in correcting V, also used another MS. to which 



1 



54 INTRODUCTION [ch, v 

he alludes in the additions to ix, 19 and 30 (and also on X, 23 For); 
"in the book of the Ephesian (this) is not found*' Who this Ephesian 
of the 1 2th c was, we do not know. 

We now come to the alterations made by Theon in his edition of 
the Elements. I shall indicate classes into which these alterations 
may be divided but without details (except in cases where they affect 
the mathematical contmt as distinct from form or language pure and 
simple)*. 

I. Alterations made by Theon where he founds or thought kt founds 
mistakes in the original 

1. Real blots in the original which Theon saw and tried to 
remove. 

{a) Euclid has a porism (corollary) to vi. 19, the enunciation 
of which speaks of similar and similarly described figures though the 
proposition itself refers only to triangles, and therefore the porism 
should have come after VI. 20. Theon substitutes triangle for figtirt 
and proves the inore general porism after VL 20. 

Qf) In IX. 19 there is a statement which is obviously incorrect. 
Theon saw this and altered the proof by reducing four alternatives to 
two, with the result that it fails to correspond to the enunciation even 
with Theon's substitution of** if*' for '* when " in the enunciation, 

{c) Theon omits a porism to IX* 11, although it is necessary for 
the proof of the succeeding proposition, apparently because, owing to 
an error in the text {Kmrk ^qv corrected by Heiberg into hti th)^ he 
could not get out of it the right sense. 

(rf) I should also put into this categoiy a case which Heiberg 
classifies among those in which Th^:^n merely fancied that he found 
mistakes, viz. the porism to V. 7 stating that, if four magnitudes are 
proportional, they are proportional inversely. Theon puts this after 
V. 4 with a proof, which however has no necessary connexion with 
V. 4 but is obvious from the definition of proportion, 

{f) I should also put under this head Xl. i, where Euclid s argu- 
ment to prove that two straight lines cannot have a common segment 
is altered. 

2. Passages which seemed to Theon to contain blots, and which 
he therefore set himself to correct, though more careful consideration 
would have shown that Euclid's words are right or at least may be 
excused and offer no difficulty to an intelligent reader. Under this 
head come : 

(a) an alteration in ill. 24. 

\b) a perfectly unnecessary alteration, in vi. 14, of "equiangular 
parallelograms" into "parallelograms having one angle equal to one 
angle,'* where Theon followed the false analogy of vi. 1 5. 

{c) an omission of words in v. 26, owing to his having been mis- 
led by a wrong figure. 

(rf) an alteration of the order of XI. Deff. 27, 28. 

\e) the substitution of " parallelepipedal solid " for " cube *' in XI. 

^ Exhaustive detaUs mider aU the difleient heads are given by Heibeig (VoL v. 
pp. lii— bov). 






CH. v] THE TEXT 55 

38, because Theon observed, correctly enough, that it was true of the 
parallelepipedal solid in general as well as of the cube, but failed to 
give weight to the fact that Euclid must have given the particular 
case of the cube for the simple reason that that was all he wanted for 
use in xill. 17. 

(/) the substitution of the letter ^ for ft ( F for Z in my figure) 
because he saw that the perpendicular from K to B^ would fall on ^ 
itself, so that ^, ft coincide. But, if the substitution is made, it should 
ht proved that 4>, ft coincide. Euclid can hardly have failed to notice 
the fact, but it may be that he deliberately ignored it as unnecessary 
for his purpose, because he did not want to lengthen his proposition 
by giving the proof 

II. Emendations intended to improve the form or diction of Euclid. 
Some of these emendations of Theon affect passages of appreciable 

length. Heiberg notes about ten such passages ; the longest is 
^ in Eucl. XII. 4 where a whole page of Heiberg's text is affected and 
I Theon's version is put in the Appendix. The kind of alteration may 
be illustrated by that in ix. 15 where Euclid uses successively the 
propositions Vll. 24, 25, quoting the enunciation of the former but not 
of the latter ; Theon does exactly the reverse. In a few of the cases 
here quoted by Heiberg, Theon shortened the original somewhat. 

But, as a rule, the emendations affect only a few words in each 
sentence. Sometimes they are considerable enough to alter the con- 
formation of the sentence, sometimes they are trifling alterations 
"more magistellorum ineptorum" and unworthy of Theon. Generally 
speaking, they were prompted by a desire to change anything which 
was out of the common in expression or in form, in order to reduce 
the language to one and the same standard or norm. Thus Theon 
changed the order of words, substituted one word for another where 
the latter was used in a sense unusual with Euclid (e.g. iweiiiprcp, 
"since," for Sn in the sense of "because"), or one expression for 
another in like circumstances (e.g. where, finding ''that which was 
enjoined would be done" in a theorem, Vll. 31, and deeming the phrase 
more appropriate to a problem, he substituted for it " that which is 
sought would be manifest"; probably also and for similar reasons he 
made certain variations between the two expressions usual at the end 
of propositions oirtp Ihei Sel^ai and Swep eSet iroifjaai, quod erat 
demonstrandum and quod erat faciendum). Sometimes his alterations 
show carelessness in the use of technical terms, as when he uses 
iiTTcaOai (to meet) for i^dtrr^cBai (to touch) although the ancients 
carefully distinguished the two words. The desire of keeping to a 
standard phraseology also led Theon to omit or add words in a 
number of cases, and also, sometimes, to change the lettering of 
figures. 

But Theon seems, in editing the Elements, to have bestowed the 
most attention upon 

III. Additions designed to supplement or explain Euclid, 
First, he did not hesitate to interpolate whole propositions where 

be thought there was room or use for them. We have already 



56 INTRODUCTION 

mentioned the addition to vi. 33 of the second part relating to sectors^ 
for which Theon himself takes credit in his commentary on Ptolemy. 
Again, he interpolated the proposition commoniy known as VI L 22 
{ex aequo in propartione perturbata for numbers, corresponding to V, 23X 
and perhaps also vil 20, a particular case of viL 19 as VL 17 is of VL 
16. He added a second case to VL 27, a porism to IL 4, a second 
porism to III. 16, and a lemma after X. 12; perhaps also the porism 
to V. 19 and the first porism to vi, 2a He also inserted alternative 
proofs here and there, e.g, in IL 4 (where the alternative differs little 
from the original) and in vn. 31 ; perhaps also in X. i, 6, and 9* 

Secondly, he sometimes repeats an argument where Euclid had 
said "For the same reason/* adds specific references to points, 
straight lines etc. in the figures in order to exclude the possibility 
of mistake arising from Euclid's reference to them in general terms, 
or inserts words to make the meaning of Euclid more plain, e.g. 
componendo and alternately^ where Euclid had left them out. Some- 
times he thought to increase by his additions the mathematical 
precision of Euclid^s language in enunciations or elsewhere, sometimes 
to make smoother and clearer things which Euclid had expressed 
with unusual brevity and harshness or carelessness, in reliance on the 
intelligence of his readers* 

Thirdly, he supplied intermediate steps where EucHd^s argument 
seemed too rapid and not easy enough to follow. The form of these 
additions varies; they are sometimes placed as a definite intermediate 
step with "therefore*' or "so that" sometimes they are additions to 
the statement of premisses, sometimes phrases introduced by ** since " 
"for" and the like, after the inference. 

Lastly, there Is a very large class of additions of a word, or one 
or two words, for the sake of clearness or consistency. Heiberg 
gives a number of examples of the addition of such nouns as 
"triangle," "square," ''rectangle,** "magnitude/* "number" "pointy" 
"side," "circle," "straight line" "area" and the like, of adjectives 
such as " remaining/' " right," " whole/' " proportional/' and of other 
parts of speech, even down to words like " is " {i<irl) which is added 
600 times. Si;, &pa^ ^ikp, ydp, scat and the like. 

IV. Omissions by Tksan. 

Heibei^ remarks that, Theon *s object having been, as above 
shown, to amplify and explain Euclid, we should not naturally have 
expected to find him doing much in the contrary process of com- 
pression, and it is only owing to the recurrence of a certain sort of 
omissions so frequently (especially in the first Books) as to exclude 
the hypothesis of their being all due to chance that we are bound to 
credit him with alterations making for greater brevity. We have 
seen, it is true, that he made omissions as well as additions for the 
purpose of reducing the language to a certain standard form. But 
there are also a good number of cases where in the enunciation of 
propositions, and in the exposition (the re-statement of them with 
reference to the figure), he has left out words because, apparently, 
he regarded Euclid's language as being too careful and precise. 



CH.V] THE TEXT 57 

Again, he is apparently responsible for the frequent omission of the 
words iir^p ioei Bel^i (or 'n'oi,fiaa4)t Q.E.D. (or F.), at the end of 
propositions. This is often the case at the end of porisms, where, 
in omitting the words, Theon seems to have deliberately departed 
from Euclid's practice. The MS. P seems to show clearly that, where 
Euclid put a porism at the end of a proposition, he omitted the 
Q.E.D. at the end of the proposition but inserted it at the end of the 
porism, as if he regarded die latter as being actually a part of the 
proposition itself. As in the Theonine MSS. the Q.E.D. is generally 
omitted, the omission would seem to have been due to Theon. 
Sometimes in these cases the Q.E.D. is interpolated at the end of the 
proposition. 

Heiberg summed up the discussion of Theon's edition by the 
remark that Theon evidently took no pains to discover and restore 
from MSS. the actual words which Euclid had written, but aimed 
much more at removing difficulties that might be felt by learners 
in studying the book. His edition is therefore not to be compared 
with the editions of the Alexandrine grammarians, but rather with 
the work done by Eutocius in editing Apollonius and with an 
interpolated recension of some of the works of Archimedes by a 
certain Byzantine, Theon occupying a position midway between these 
two editors, being superior to the latter in mathematical knowledge 
but behind Eutocius in industry (these views now require to be some- 
what modified, as above stated). But however little Theon's object 
may be approved by those of us who would rather know the 
ipsissima verba of Euclid, there is no doubt that his work was 
approved by his pupils at Alexandria for whom it was written ; and 
his edition was almost exclusively used by later Greeks, with the 
result that the more ancient text is only preserved to us in one MS. 

As the result of the above investigation, we may feel satisfied 
that, where P and the Theonine MSS. agree, they give us (except in a 
few accidental instances) Euclid as he was read by the Greeks of 
the 4th c. But even at that time the text had been passed from 
hand to hand through more than six centuries, so that it is certain 
that it had already suffered changes, due partly to the fault of 
copyists and partly to the interpolations of mathematicians. Some 
errors of copyists escaped Theon and were corrected in some MSS. 
by later hands. Others appear in all our MSS. and, as they cannot 
have arisen accidentally in all, we must put them down to a common 
source more ancient than Theon. A somewhat serious instance is 
to be found in III. 8 ; and the use of amriadta for i^irriaOta in the 
sense of "touch" may also be mentioned, the proper distinction 
between the words having been ignored as it was by Theon also. 
But there are a number of imperfections in the ante-Theonine text 
which it would be unsafe to put down to the errors of copyists, those 
namely where the good MSS. agree and it is not possible to see any 
motive that a copyist could have had for altering a correct reading. 
In these cases it is possible that the imperfections are due to a 
certain degree of carelessness on the part of Euclid himself; for it 



S8 INTRODUCTION [ch. v 

is not p(^sib1e ** Euclidem ab omni naevo vindicare/' to use the 
words of SaccheriSand consequently Simson ts not right in attributing 
to Theon and other editors all the things in EucHd to which mathe- 
matical objection can be taken. Thiis^ when EucHd speaks of "the 
ratio compounded of the sides" for *'the ratio compounded of the 
ratios of the sides/* there is no reason for doubting that Euclid himself 
is responsible for the more slip-shod expression. Again, in the Books 
XI. — XIIL relating to solid geometry there arc blots neither few 
nor altc^ether unimportant which can only be attributed to Euclid 
himself*; and there is the less reason for hesitation in so attributing 
them because solid geometry was then being treated in a thoroughly 
systematic manner for the first time. Sometimes the cofulusion 
{cvfvrripcLciia) of a proposition does not correspond exactly to the 
enunciation, often it is cut short with the w^ords koX ra efij? " and the 
rest" (especially from Book X. onwards), and very often in Books VIII*, 
IX. it is omitted. Where all the MSS, agree, there is no ground for 
hesitating to attribute the abbreviation or omission to Euclid ; though, 
of course, where one or more Ms*>. have the longer form, it must be 
retained because this is one of the cases where a copyist has a 
temptation to abbreviate. 

Where the true reading is preserved in one of the Theonine MSS, 
alone, Heibcrg attributes the wrong reading to a mistake which arose 
before Theon's time, and the right reading of the single MS, to a 
successful correction* 

We now come to the most important question of the Interpolations 
introduced befare Tkeatis time. 

I. Alternative proofs or additional cases* 

It is not in itself probable that Euclid would have given two 
proofs of the same proposition ; and the doubt as to the genuineness 
of the alternatives is increased when we consider the character of 
some of them and the way in which they are introduced. First of 
all, we have those of VL 20 and xn. 17 introduced by '*we shall prove 
this otherwise more readily (wpox^ipojef^opy* or that of X, 90 " it is 
possible to prove mare slwrtly (<yvifrofiwr€popy Now it is impossible 
to suppose that Euclid would have given one proof as that definitely 
accepted by him and then added another with the express comment 
that the latter has certain advantages over the former. Had he con- 
sidered the two proofs and come to this conclusion, he would have 
inserted the latter in the received text instead of the former. These 
alternative proofs must therefore have been interpolated. The same 
argimient applies to alternatives introduced with the words " or even 
thus" (fj «cai o{hrfl09)> ''or even otherwise" (tf teal aXXo^). Under this 
head come the alternatives for the last portions of IIL 7, 8 ; and 
Heibcrg also compares the alternatives for parts of IIL 31 (that the 
angle in a semicircle is a right angle) and xill. 18, and the alternative 
proof of the lemma after x. 32. The alternatives to x. 105 and 106, 

^ EmcUda ab cmm uoivo vindUatus^ Mediolani, T733. 

* Cf. espediUy the anomption, without proof or definition, of the criterion for eqtml solid 
aa^et, and the incomplete proof of xn. 17. 



I CH. V 

f aeaii 



CH. v] THE TEXT 59 

again, are condemned by the place in which they occur, namely after 
an alternative proof to X. 115. The above alternatives being all 
admitted to be spurious, suspicion must necessarily attach to the few 
others which are in themselves unobjectionable. Heibei^ instances 
the alternative proofs to III. 9, III. 10, VI. 30, VI. 3 1 and XI. 22, observing 
that it is quite comprehensible that any of these might have occurred 
to a teacher or editor and seemed to him, rightly or wrongly, to be 
better than the corresponding proofs in Euclid. Curiously enough, 
Simson adopted the alternatives to HI. 9, 10 in preference to the 
genuine proofs. Since Heiberg's preface was written, his suspicion 
has been amply confirmed as regards III. 10 by the commentary of 
an-NairlzI (ed. Curtze) which shows not only that this alternative is 
Heron's, but also that the substantive proposition III. 12 in Euclid 
is also Heron's, having been given by him to supplement III. 11 
which must originally have been enunciated of circles " touching one 
another" simply, i.e. so as to include the case of external as well as 
internal contact, though the proof covered the case of internal contact 
only. "Euclid, in the iith proposition," says Heron, "supposed two 
circles touching one another internally and wrote the proposition on 
this case, proving what it was required to prove in it But I will 
show haw it is to be proved if the contact be extemalK** This additional 
proposition of Heron's is by way of adding another case^ which brings 
us to that class of interpolation. It was the practice of Euclid and 
the ancients to give only one case (generally the most difficult one) 
and to leave the others to be investigated by the reader for himself. 
One interpolation of a second case (vi. 27) is due, as we have seen, 
to Theon. The two extra cases of XI. 23 were manifestly interpolated 
before Theon's time, for the preliminary distinction of three cases, 
** (the centre) will either be within the triangle LMN^ or on one of 
the sides, or outside. First let it be within," is a spurious addition 
(B and V only). Similarly an unnecessary case is interpolated in 
III. II. 

H. Lemmas. 

Heiberg has unhesitatingly placed in his Appendix to Vol. III. 
certain lemmas interpolated either by Theon (on X. 13) or later 
writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas). 
But we are here concerned with the lemmas found in all the MSS., 
which however are, for different reasons, necessarily suspected. We 
will deal with the Book X. lemmas last 

(i) There is an a priori ground of objection to those lemmas 
which come after the propositions to which they relate and prove 
properties used in those propositions ; for, if genuine, they would be a 
sign of faulty arrangement such as would not be likely in a systematic 
work so carefully ordered as the Elements. The lemma to VI. 22 is 
one of this class, and there is the further objection to it that in VI. 28 
Euclid makes an assumption which would equally require a lemma 
though none is found. The lemma after XII. 4 is open to the further 
objections that certain altitudes are used but are not drawn in the 

^ An-Nairld, ed. Curtze, p. isi. 




6o ^ INTRODUCTION [cH. v 

figure (which is not in the manner of Euclid), and that a peculiar 
expression ** parallele pi pedal solids described on (apaypatf^ofieva aw6) 
pristns** betrays a hand other than Euclid's. There is an objection on 
the score of language to the lemma after xni. 2. The lemmas on 
XL 23, XIII. 13, XIIL 18, besides coming after the propositions to 
which they relate, are not very necessary in themselves and, as regards 
the lemma to XIIL 13, it is to be noticed that the writer of a gloss 
in the proposition could not have had it, and the words "as will 
be proved afterwards** in the text are rightly suspected owing to 
differences between the MS, readings. The lemma to Xll* 2 also, to 
which Simson raised objection, comes after the proposition ; but, if it 
is rejected, the words ** as was proved before" used in XIL 5 and i8^ 
and referring to this lemma, must be struck out 

(2) Reasons of substance are fatal to the lemma before X 60, 
which is really assumed in X. 44 and therefore should have appeared 
there if anywhere, and to the lemma on X. 20, which tries to prove 
what is already stated in X. Det 4. 

We now come to the remaining lemmas in Book X., eleven in 
number, which come before the propositions to which they relate and 
remove difficulties in the way of their demonstration. That before 
X, 42 introduces a set of propositions with the words *' that the said 
irrational straight lines are uniquely divided >.,we will prove after 
premising the following lemma," and it is not possible to suppose 
that these words are due to an interpolator ; nor are there any 
objections to the lemmas before x, 14, 17^ 22, 33, 54, except perhaps 
that they are rather easy. The lemma before X. 10 and X io itself 
should probably be removed from the Elements ; for X, iO really uses 
the following proposition X. il, which is moreover numbered 10 by 
the firsthand in P, and the words in X, lo referring to the lemma "for 
we learnt (how to do this) '* betray the interpolator Heiberg gives 
reason also for rejecting the lemmas before X. 19 and 24 with the 
words "in any of the aforesaid ways " (omitted in the Theonine MSS.) 
in the enunciations of x. 19, 24 and in the exposition of X. 20. Lastly, 
the lemmas before x, 29 may be genuine, though there is an addition 
to the second of them which is spurious. 

Heiberg includes under this heading of interpolated lemmas two 
which purport to be substantive propositions, XL 38 and XIIL 6, These 
must be rejected as spurious for reasons which will be found in detail 
in my notes on XL 37 and xml 6 respectively. The latter proposition 
is only quoted once (in XIii. 17); probably the words quoting it 
(with 7/9a/i/Ai^ instead of eiOeia) are themselves interpolated, and 
Euclid thought the fact stated a sufficiently obvious inference from 
Xlll. I. 

III. Porisms (or corollaries). 

Most of the porisms in the text are both genuine and necessary ; 
but some are shown by differences in the MSS. not to be so, e.g. those 
to L 15 (though Proclus has it), III. 31 and VL 20 (For. 2). Sometimes 
parts of porisms are interpolated. Such are the last few lines in 
tlie porisms to iv. 5, yL 8 ; the latter addition is proved later by 



CH. v] THE TEXT 6i 

means of vi. 4, 8, so that the writer of these proofs could not have had 
the addition to vi. 8 For. before him. Lastly, interpolators have added 
a sort of proof to some porisms, as though they were not quite 
obvious enough ; but to add a demonstration is inconsistent with the 
idea of a porism, which, according to Proclus, is a by-product of a 
proposition appearing without our seeking it 
IV. Scholia. 

Several interpolated scholia betray themselves by their wording, 
e.g. those given by Heibei^ in the Appendix to Book X. and contain- 
ing the words KoKel^ iKoKeat ("he calls" or "called"); these scholia were 
apparently written as marginal notes before Theon's time, and, being 
I adopted as such by Theon, found their way into the text in P and 
I some of the Theonine MSS. The same thing no doubt accounts for 
I the interpolated analyses and syntheses to xiil. i — 5, as to which see 
\ my note on xill. i. 
f V. Interpolations in Book X. 

First comes the proposition " Let it be proposed to us to show that 
in square figures the diameter is incommensurable in length with the 
side," which, with a scholium after it, ends the tenth Book. The form 
of the enunciation is suspicious enough and the proposition, the proof 
of which is indicated by Aristotle and perhaps was Pythagorean, is 
perfectly unnecessary when X. 9 has preceded. The scholium ends 
with remarks about commensurable and incommensurable solids, 
which are of course out of place before the Books on solids. The 
scholiast on Book x. alludes to this particular scholium as being due 
to " Theon and some others." But it is doubtless much more ancient, 
and may, as Heiberg conjectures, have been the b^inning of 
Apollonius' more advanced treatise on incommensurables. Not only 
is everything in Book X. after X. 1 15 interpolated, but Heiberg doubts 
the genuineness even of x. 112 — 115, on the ground that x. in 
rounds ofT the theory of incommensurables as we want it in the Books 
on solid geometry, while x. 112 — 115 are not really connected with 
what precedes, nor wanted for the later Books, but seem to form the 
starting-point of a new and more elaborate theory of irrationals. 

VI. Other minor interpolations are found of the same character as 
those above attributed to Theon. First there are two places (XL 35 
and XI. 26) where, after "similarly we shall prove " and "for the same 
reason," an actual proof is nevertheless given. Clearly the proofs are 
interpolated; and there are other similar interpolations. There 
are also interpolations of intermediate steps in proofs, unnecessary 
explanations and so on, as to which I need not enter into details. 

Lastly, following Heiberg's order, I come to 

VII. Interpolated definitions, axioms etc. 
Apart from VI. Def. 5 (which may have been interpolated by 

Theon although it is found written in the margin of P by the first 
hand), the definition of a segment of a circle in Book I. is interpolated, 
as is clear from the fact that it occurs in a more appropriate place in 
Book III. and Proclus omits it. \l. Def. 2 (reciprocal figures) is rightly 
condemned by Simson — perhaps it was taken from Heron — and 



i. 



i 



C9 INTRODUCTION [ch, v 

Heiber^ would reject VI I, Def lo, as to which see my note on that 
definition. Lastly the double definition of a solid angle (XL Def ii) 
constitutes a difficulty. The use of the word iTrt^idvtta suggests that 
the first definition may have been older than Euclid, and he may have 
quoted it from older iiemtnts^ especially as his own definition which 
follows only includes solid angles contained by planes, whereas the m, 
other includes other sorts (cf the words ^pap^fio^v, ypafi^<;) which are II 
also distinguished by Heron (Def. 24). If the first definition had i 
come last, it could have been rejected without hesitation : but it is not 1 
so easy to reject the first part up to and including "otherwise" 
{aXXm<;). No difficulty need be felt about the definitions of '* oblong," 
"rhombus" "trapezium" and "rhomboid" which are not actually 
used in the Elematis; they were no doubt taken from earlier dements 
and given for the sake of completeness. 

As regards the axioms or, as they are called in the text, common 
notions {ttmvai ivvotai), it is to be observed that Proclus says* that 
Apollonius tried to prove ''the axioms/' and he gives Apollonius* 
attempt to prove Axiom k This shows at all events that Apollonius 
had some of the axioms now appearing in the text. But how could 
Apollonius have taken a controversial line against Euclid on the 
subject of axioms if these axioms had not been Euclid's to his know- 
ledge? Andi if they had been interpolated between Euclid's time 
and his own, how could Apollonius, living so comparatively short a 
time after Euclid, have been ignorant of the fact ? Therefore some of 
the axioms are Euclid's (whether he called them common notions, or 
axioms^ as is perhaps more likely since Proclus calls them axioms): 
and we need not hesitate to accept as genuine the first three discussed 
by Proclus, viz. (i) things equal to the same equal to one another, 
(2) if equals be added to equals, wholes equal, (5) if equals be 
subtracted from equals, remainders equal. The other two mentioned 
by Proclus (whole greater than part, and congruent figures equal) are 
more doubtful, since they are omitted by Heron, Martianus Capella, 
and others. The axiom that "two lines cannot enclose a space" is 
however clearly an interpolation due to the fact that I. 4 appeared to 
require it The others about equals added to unequals, doubles of 
the same thing, and halves of the same thing are also interpolated ; 
they are connected with other interpolations, and Proclus clearly 
used some source which did not contain them. 

Euclid evidently limited his formal axioms to those which seemed 
to him most essential and of the widest application ; for he not un- 
frequently assumes other things as axiomatic, e.g. in vil. 28 that, if a 
number measures two numbers, it measures their difference. 

The differences of reading appearing in Proclus suggest the 

Question of the comparative purity of the sources used by Proclus, 
leron and others, and of our text The omission of the definition of 
a s^ment in Book I. and of the old gloss ** which is called the cir- 
cumference" in L Def. 15 (also omitted by Heron, Taurus, Sextus 

* Pkoehu, pp. 194, loaqq. 



CH. v] THE TEXT 63 

Empiricus and others) indicates that Proclus had better sources than 
we have ; and Heibci^ gives other cases where Proclus omits words 
which are in all our MSS. and where Proclus' reading should perhaps 
I be preferred. But, except in these instances (where Proclus may have 
drawn from some ancient source such as one of the older com- 
mentaries), Proclus' MS. does not seem to have been among the best 
Often it agrees with our worst MSS., sometimes it agrees with F where 
F alone has a certain reading in the text, so that (e.g. in I. 15 Por.) 
the common reading of Proclus and F must be rejected, thrice only 
does it agree with P alone, sometimes it agrees with P and some 
Theonine MSS., and once it agrees with the Theonine MSS. against P 
and other sources. 

Of the other external sources, those which are older than Theon 
generally agree with our best MSS., e.g. Heron, allowing for the 
difference in the plan of his definitions and the somewhat free adap- 
tation to his purpose of the Euclidean definitions in Books X., XI. 

Heiberg concludes that the Elements were most spoiled by inter- 
polations about the 3rd c, for Sextus Empiricus had a correct text, 
while lamblichus had an interpolated one; but doubtless the purer 
text continued for a long time in circulation, as we conclude from the 
fact that our MSS. are free from interpolations already found in 
lamblichus' MS. 







CHAPTER VL 

THE SCHOLIA- 

» ^ t , 

Heiberg has collected scholia, to the number of about 1500, in 
Vol V* of his edition of EucHd, and has also discussed and classified 
them in a separate short treatise, in which he added a few others*. 

These scholia cannot be regarded as doing much to facilitate the 
reading of the Eitntents. As a rule, they contain only such observa- 
tions as any intelligent reader could make for himself Among the 
few exceptions are XL Nos. 33, 35 (where XL 22^ 23 are extended to 
solid angles formed by any number of plane angles), XI L No. 85 
(where an assumption tacitly made by Euclid in XIL 17 is proved), 
IX. Nos, 28, 29 (where the scholiast has pointed out the error in the 
text of JX. 19). 

Nor are they very rich in historical information ; they cannot be 
compared in this respect with Proclus' commentary on Book I* or 
with those of Eutocius on Archimedes and Apollonius, But even 
tinder this head they contain some things of interest, e.g. IL No. 1 1 
explaining that the gnomon was invented by geometers for the sake of 
brevity, and that its name was suggested by an incidental characteristic^ 
namely that "from it the whole is known {jvmpi^^rai), ^Mh^t of the 
whole area or of the remainder, when it (the ywfL^v) is either placed 
round or taken away"; iL No. 13, also on the gnomon; IV, No. 2 
stating that Book IV. was the discovery of the Pythagoreans ; 
V, Na I attributing the content of Book v, to Eudoxus; X. No. i with 
its allusion to the discovery of incommensurability by the Pytha- 
goreans and to Apollonius* work on irrationals ; X. No. 62 definitely 
attributing X, 9 to Theaetetus; XIIL No. i about the "Platonic" figures, 
which attributes the cube, the pyramid, and the dodecahedron to the 
Pythagoreans, and the octahedron and icosahedron to Theaetetus. 

Sometimes the scholia are useful in connexion with the settlement 
of the text, (i) directly^ e.g, IIL Na 16 on the interpolation of the 
word "within'* (iirri^) in the enunciation of IIL 6, and X, No. 1 
alluding to the discussion by "Theon and some others" of irrational 
"surfaces" and '* solids," as well as '*lines/' from which we may 

^ Heiberg, Om Stka/vrm tU EttkUth Elrminitr^ Kjifbenhft^Q, |S8S, Tbe tr^ct is 
wniten is Dankh, but, fortuniLt^ly for those who do not read Danish easilj, Lhe author has 
ftprpended (pp. 7^—78} a re&mne in French* 






CH.VI] THE SCHOLIA 65 

conclude that the scholium at the end of Book x/ is not genuine ; 

(2) indirectly in that they sometimes throw light on the connexion 
of certain MSS. 

Lastly, they have their historical importance as enabling us to 
judge of the state of mathematical science at the times when they 
were written. 

Before passing to the classification of the scholia, Heiberg remarks 
that we must separate from them a number of additions in the nature 
of scholia which are found in the text of our MSS. but which can, in 
one way or another, be proved to be spurious. As they are found 
both in P and in the Theonine MSS., they must have been in the MSS. 
anterior to Theon (4th c). But they are, in great part, only found in 
the margin of P and the Theonine MSS.; in V they are half in the 
text and half in the margin. This can hardly be explained except 
on the supposition that these additions were originally (in the MSS. 
before Theon) in the margin, and that Theon kept them there in his 
edition, but that they afterwards found their way gradually into the 
text of P as well as of the Theonine MSS., or were omitted altogether, 
while particular MSS. have in certain places preserved the old arrange- 
ment Of such spurious additions Heiberg enumerates the following: 
the axiom about equals subtracted from unequals, the last lines of the 
porism to VI. 8, second porisms to V. 19 and to vi. 20, the porism 
to III. 31, VI. Def. 5, various additions in Book X., the analyses and 
syntheses of XIII. i — 5, and the proposition XIII. 6. 

The two first classes of scholia distinguished by Heiberg are 
denoted by the convenient abbreviations "SchoL Vat" and "Schol. 
Vind." 

L Schol. Vat. 

It is first necessary to set out the letters by which Heiberg 
denotes certain collections of scholia. 

P « Scholia in P written by the first hand. 

B a Scholia in B by a hand of the same date as the MS. itself, 
generally that of Arethas. 

F« Scholia in F by the first hand. 

Vat = Scholia of the Vatican MS. 204 of the loth c, which has 
these scholia on leaves 198 — 205 (the end is missing) as an independent 
collection. It does not contain the text of the Ekments, 

V*^ = Scholia found on leaves 283 — 292 of V and written in the 
same hand as that part of the MS. itself which begins at leaf 235. 

Vat 192 =s a Vatican MS. of the 14th c which contains, after 
(i) the Elements I. — XIli. (without scholia), (2) the Data with scholia, 

(3) Marinus on the Data^ the Schol. Vat as an independent collection 
and in their entirety, b^inning with I. No. 88 and ending with Xlll. 
No. 44. 

The Schol. Vat, the most ancient and important collection of 

scholia, comprise those which are found in PBF Vat. and, from vil. 12 

. to IX. 1 5, in PB Vat. only, since in that portion of the Elements 

F was restored by a later hand without scholia ; they abo include I. 

H. £. e . 



66 INTRODUCTION [ch. vi 

No. 88 which onfy happens to be erased in F, and IX. Nos, 28, 29 
which may be left out because F here has a different text In F 
and Vat the collection ends with Book X, ; but it must also include 
SchoL PB of Books XL^ — xilt, since these are found along with Schol* 
Vat to Books L— X* in several MSS. (of which Vat 192 is one) as a 
separate collection. The SchoL Vat to Books X,— XIIL are also 
found in the collection V^ (where, curiously enough, Xllt Nos. 45, 44 
are at the beginning). The Scho!, Vat. accordingly include SchoL 
PBV*= Vat 193, and doubtless also those which are found in two of 
these sources. The total number of scholia classified by Heiberg as 
SchoL Vat is 138, 

As r^ards the contents of SchoL Vat Heiberg has the following 
observations. The thirteen scholia to Book T. are extracts made 
from Proclus by a writer thoroughly conversant with the subject, 
and cleverly recast (with some additions). Their author does not 
seem to have had the two lacunae which our text of Proclus has 
(at the end of the note on t 36 and the beginning of the next note^ 
and at the beginning of the note on L 43), for the scholia L Nos. 125 
and 137 seem to hll the gaps appropriately, at least in part. In 
some passages he had better readings than our MSS. have. The rest 
of SchoL Vat (on Books ll. — XIII.) are essentially of the same 
character as those on Book I., containing prolegomena, remarks on 
the object of the propositions, critical remarks on the text, converses, 
lemmas; they are, in general, exact and true to tradition. The 
reason of the resemblance between them and Proclus appears to be 
due to the fact that they have their origin in the commentary of 
Pappus, of which we know that Proclus also made use. In support 
of the view that Pappus is the source, Heiberg places some of the 
SchoL Vat to Book x, side by side with passages from the com- 
mentary of Pappus in the Arabic translation discovered by Woepcke* ; 
he also refers to the striking confirmation afiforded by the fact that 
XIL No. 2 contains the solution of the problem of inscribing in a 
given circle a polygon similar to a polygon inscribed in another circle, 
which problem Eutocius says' that Pappus gave in his commentary 
on the Ekmettts. 

But, on the other hand, SchoL Vat. contain some things which 
cannot have come from Pappus, e.g. the allusion in X. No, i to Theon 
and irrational surfaces and solids, Theon being later than Pappus ; 
lit No, 10 about porisms is more like Proclus' treatment of the 
subject than Pappus', though one expression recalls that of Pappus 
about/ormtn^ (axnM^ri^eaOai) the enunciations of porisms like those 
of either theorems or problems. 

The Schol. Vat give us important indications as regards the 
text of the Elements as Pappus had it In particular, they show that 
he could not have had in his text certain of the lemmas in Book x. 
For example, three of these are identical with what we find in Schol. 

^ Om ScMitrm tU Euklids Elenunttr^ pp. 11, 13 : cf. EukHd'Stutiien^ pp. 170, 171 ; 
Woepcke, Mimciru friunt. d tAcad, da Scufues, 1856, xiv. p. 658 sqq. 
* Archimedes, ed. Hdbeig, iiL p. 54, 5 — 8. 



I 



J 






(| 

; I CH. VI] THE SCHOLIA 67 

Vat (the lemma to x. i7 = Schol. x. No. 106, and the lemmas to 
• X. 54, 60 come in Schol. X. No. 328) ; and it is not possible to suppose 
* \ that these lemmas, if they were already in the text, would also be 
given as scholia. Of these three lemmas, that before X. 60 has 
already been condemned for other reasons; the other two, un- 
objectionable in themselves, must be rejected on the ground now 
stated. There were four others against which Heiberg found nothing 
to urge when writing his prolegomena to Vol. v., viz. the lemmas 
before X. 42, x. 14, x. 22 and X. 33. Of these, the lemma to X. 22 
is not reconcilable with Schol. X. No. 161, which takes up the 
assumption in the text of Eucl. x. 22 as if no lemma had gone before. 
The lemma to X. 42, which, on account of the words introducing it 
(see p. 60 above), Heiberg at first hesitated to regard as an inter- 
polation, is identical with Schol. X. No. 270. It is true that in 
Schol. X. No. 269 we find the words "this lemma has been proved 
before (iv roh ifiirpoaOev), but it shall also be proved now for 
convenience' sake (rov iroifiov tv€KaY and it is possible to suppose 
that " before " may mean in Euclid's text before X. 42 ; but a proof 
in that place would surely have been as ** convenient " as could be 
desired, and it is therefore more probable that the proof had been 
given by Pappus in some earlier place. (It may be added that the 
lemma to x. 14, which is identical with the lemma to XI. 23, con- 
demned on other grounds, is for that reason open to suspicion.) 

Heiberg's conclusion is that all the lemmas are spurious, and that 
most or all of them have found their way into the text from Pappus' 
commentary, though at a time anterior to Theon's edition, since 
they are found in all our MSS. This enables us to fix a date for these 
interpolations, namely the first half of the 4th c. 

Of course Pappus had not in his text the interpolations which, 
from the fact of their appearing only in some of our MSS., are seen to 
be later than those above-mentioned. Such are the lemmas which 
are found in the text of V only after x. 29 and x. 31 respectively and 
are given in Heiberg's Appendix to Book x. (numbered 10 and 11). 
On the other hand it appears from Woepcke's tract^ that Pappus 
already had X. 115 in his text : though it does not follow from this 
that the proposition is genuine but only that interpolations began 
very early. 

Theon interpolated a proposition (or lemma) between X. 12 and 
X. 13 (No. 5 in Heiberg*s Appendix). Schol. Vat. has the same 
thing (X. No. 125). The writer of the scholia therefore did not find 
this lemma in the text. Schol. Vat. IX. Nos. 28, 29 show that neither 
did he find in his text the alterations which Theon made in Eucl. IX. 
19; the scholia in fact only agree with the text of P, not with Theon's. 
This suggests that Schol. Vat. were written for use with a MS. of the 
ante-Theonine recension such as P is. This probability is further 
confirmed by a certain independence which P shows in several places 
when compared with the Theonine MSS. Not only has P better 
readings in some passages, but more substantial divergences, and, 

* Woepcke, op, cit. p. 70a. 

5—2 



».i. 



» introi>ucti6n 

in particularp the absence in P of three notes of a historical character 
which are added ^ wholly or partly from Proclus, in the Theonine MSS, 
attests an independent and more primitive point of view in P, 

In view of the distinctive character of P, it is possible that some 
of the scholia found in it in the first hand, but not in the other 
sources of SchoK Vat, also belong to that collection ; and several 
circumstances confirm this. Schol, XI I L No. 45. found in P only, 
which relates to a passage in Eucl Xill. 15, shows that certain words 
in the text, though older than Theon» are interpolated ; and, as the 
scholium is itself older than Theon, is headed *' tkird lemma/' and 
follows a "second lemma" relating to a passage in the text im- 
mediately preceding, which ** second lemma" belongs to Schol Vat 
and is taken from Pappus, the *'third" in all probability came from 
Pappus also. The same is true of SchoL XlL No. 72 and XII L No* 69, 
which are respectively identical with the propositions t^u/^o XL 38 
(Heiberg, App, to Book XL, No. 5) and xlll. 6; for both of these 
interpolations are older than Theon, Moreover most of the scholia 
which P in the first hand alone has are of the same character as 
Schoh Vat Thus VIL No. 7 and XIIL No. i introducing Books vil. 
and XIIL respectively are of the same historical character as several 
of SchoL Vat, ; that Vll, No. 7 appears in the t€zt of P at the 
beginning of Book vil constitutes no difficulty. There are a number 
of converses^ remarks on the relation of propositions to one another, 
explanations such as XIL No. 89 in which it is remarked that 4>, H 
in Euclid's figure to XIL 17 {Z^ V in my figure) are really the same 
point but that this makes no difference in the proof. Two other 
Schol. P on XIL 17 are connected by their headings with xn. No. 72 
mentioned above, XL No, 10 (P) is only another form of XL 
No, 1 1 (B) ; and B often, alone with P, has preserved Schol. Vat 
On the whole Heiberg considers some 40 scholia found in P alone to 
belong to Schol. Vat 

The history of Schol, Vat. appears to have been, in its main 
outlines, the following. They were put together after 500 A<D., since 
they contain extracts from Proclus, to which we ought not to assign 
a date too near to that of Proclus' work itself; and they must at least 
be earlier than the latter half of the 9th c, in which B was written. 
As there must evidently have been several intermediate links between 
the archetype and B, we must assign them rather to the first half of 
the period between the two dates, and it is not improbable that they 
were a new product of the great development of mathematical studies 
at the end of the 6th c. (Isidorus of Miletus). The author extracted 
what he found of interest in the commentary of Proclus on Book L 
and in that of Pappus on the rest of the work, and put these extracts 
in the margin of a MS, of the class of P, As there are no scholia to 
L I — 22, the first leaves of the archetype or of one of the eariiest 
copies must have been lost at an early date, and it was from that 
mutilated copy that partly P and partly a MS. of the Theonine class 
were taken, the scholia being put in the margin in both. Then the 
collection spread through the Theonine MSS., gradually losing some 



CH.VI] THE SCHOLIA 69 



I ca.Y 

I scholia which could not be read or understood, or which were 

I accidentally or deliberately omitted. Next it was extracted from 

\ one of these MSS. and made into a separate work which has been 

I preserved, in part, in its entirety (Vat 192 etc.) and, in part, divided 

I into sections, so that the scholia to Books X. — XIII. were detached 

; (V^). It had the same fate in the MSS. which kept the original 

arrangement (in the mai^in), and in consequence there are some MSS. 

where the scholia to the stereometric Books are missing, those Books 

having come to be less read in the period of decadence. It is from 

one of these MSS. that the collection was extracted as a separate work 

such as we find it in Vat. (loth c). 

II. The second great division of the scholia is SchoL Vind. 

This title is taken from the Viennese MS. (V), and the letters used 
by Heiberg to indicate the sources here in question are as follows. 

V* = scholia in V written by the same hand that copied the MS. 
itself from fol. 235 onward. 

q=s scholia of the Paris MS. 2344 (q) written by the first hand. 

1 = scholia of the Florence MS. Laurent XXVlii, 2 written in the 
13th-- 14th c, mostly in the first hand, but partly in two later 
hands. 

V^s= scholia in V written by the same hand as the first part 
(leaves i— 183) of the MS. itself; V^ wrote his scholia after V*. 

q* = scholia of the Paris MS. (q) found here and there in another 
hand of early date. 

Schol. Vind. include scholia found in VH). 1 is nearly related to 
q ; and in fact the three MSS. which, so far as Euclid's text is con- 
cerned, show no direct interdependence, are, as regards their scholia, 
derived from one original. Heiberg proves this by reference to the 
readings of the three in two passages (found in Schol. I. No. 109 and 
X. No. 39 respectively). The common source must have contained, 
besides die scholia found in the three MSS. V*ql, those also which 
are contained in two of them, for it is more unlikely that two of the 
three should contain common interpolations than that a particular 
scholium should drop out of one of them. Besides V* and q, the 
scholia V^ and q' must equally be referred to Schol. Vind., since the 
greater part of their scholia are found in 1. There is a lacuna in q 
from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements 
Schol. Vind. are represented by VI only. Heiberg pves about 450 
numbers in all as belonging to this collection. 

Schol. Vind. did not all come from one source ; this is shown by 
diflferences of substance, e.g. between x. Nos. 36 and 39, and by 
differences of time of writing : e.g. VI. No. 52 refers at the b^inning 
to No. 55 with the words "as the scholium has it" and is therefore 
later than that scholium ; X. No. 247 is also later than X. No. 246. 

The scholia to Book I. are here also extracts from Proclus, but 
more copious and more verbatim than in Schol. Vat The author 
has not always understood Proclus; and he had a text as bad as 
that of our MSS., with the same lacunae. The scholia to the other 



INTRODUCTION ^> [ch. vi 

Mil*' *■ 



3^f*; 



Books are partly drawn (r) from Schol Vat, the MSS. representing 
Schol Vind. and Schol. Vat in these cases showing nearly all possible 
combinations; but there is no certain trace in Schol Vind< of the 
scholia peculiar to P. The author used a copy of Schol Vat, in the 
form in which they were attached to the Theonine text ; thus Schol 
Vind. correspond to BF Vat, where these diverge from P, and 
especially closely to B, Besides Schol Vat., the editors of Schol 
Vind. used (2) other old collections of scholia of which we find traces 
in B and F; Schol Vind, have also some scholia common with b. 
The scholia which Schol Vind, have in common with BF come from 
two different sources, and were apparently afterwards introduced 
into the other MSS. ; one result of this is that several scholia are 
reproduced twice. 

Butj besides the scholia derived from these sources, Schol Vind 
contain a large number of others of late date, characterised by in- 
correct language or by triviality of content (there are many examples 
in numbers, citations of propositions used, absurd diropiai, and the 
hke). Unlike Schol Vat, these scholia often quote words from Euclid 
as a heading (in one case a heading is inserted in Schol Vind. where 
a scholium without the heading is quoted from Schol Vat, see V. 
No. 14). The explanations given often presuppose very little know- 
ledge on the part of the reader and frequently contain obscurities 
and gross errors. 

Schol Vind* were collected for use with a US. of the Theonine 
class; this follows from the fact that they contain a note on the 
proposition vuigo VII. 22 interpolated by Theon (given in Heiberg's 
A pp. to Vol II. p. 430). Since the scholium to vil 39 given in V and 
p in the text after the title of Book VIIL quotes the proposition as 
VII. 39, it follows that this scholium must have been written before 
the interpolation of the two propositions vu^a VII. 20, 22 ; Schol 
Vind. contain (viL No* 80) the first sentence of it, but without the 
heading referring to VI l* 39* Schol VIL No* 97 quotes vlL 33 as 
vn* 34, so that the proposition vui^ VII. 22 may have stood in the 
scholiast's text but not the later interpolation vu/go VIL 20 (later 
because only found in B in the margin by the first hand). Of course 
the scholiast had also the interpolations earlier than Theon* 

For the date of the collection we have a lower limit in the date 
(12th c.) of MSS. in which the scholia appear* That it was not much 
earlier than the 1 2th c. is indicated ( r ) by the poverty of its contents, 
(2) by the quality of the MS. of Proclus which was used in the 
compilation of it (the Munich MS. used by Friedlein with which the 
scholiast's excerpts are essentially in agreement belongs to the i ith — 
t2th c.)^ (3) by the fact that Schol Vind. appear only in uss. of the 
1 2th c* and no trace of them is found in our MSS, belonging to 
the 9th — loth c* in which Schol Vat are found. The collection may 
therefore probably be assigned to the iith c. Perhaps it may be in 
part due to Psellus who lived towards the end of that century : for in 
a Florence MS. (Magltabecch. Xl^ 53 of the 15th c) containing a 
mathematical compendium intended for use in the reading of Aristotle 






CH. VI] THE SCHOLIA 71 

the scholia I. Nos. 40 and 49 appear with the name of Psellus 
attached. 

f.* Schol. Vind. are not found without the admixture of foreign 

M elements in any of our three sources. In 1 there are only very few 
such in the first hand. In q there are several new scholia in the first 
hand, for the most part due to the copyist himself The collection of 
scholia on Book X. in q (Heiberg's q^) is also in the first hand ; it is 
not original, and it may perhaps be due to Psellus (Maglb. has some 
definitions of Book X. with a heading "scholia of... Michael Psellus 
on the definitions of Euclid's loth Element'' and Schol. x. No. 9), 
whose name must have been attached to it in the common source of 
Maglb. and q ; to a great extent it consists of extracts from Schol. 
Vind. taken from the same source as VI. The scholia q* (in an 
ancient hand in q), confined to Book II., partly belong to Schol. Vind. 
and partly correspond to W (Bologna MS.), q* and q^ are in one hand 
(Theodorus Antiochita), the nearest to the first hand of q ; they are 
doubtless due to an early possessor of the MS. of whom we know 
nothing more. 

V* has, besides Schol. Vind., a number of scholia which also appear 
in other MSS., one in BFb, some others in P, and some in v (Codex 
Vat 1038, 13th c.) ; these scholia wei-e taken from a source in which 
many abbreviations were used, as they were often misunderstood by 
V". Other scholia in V* which are hot found in the older sources — 
some appearing in V* alone — are also not original, as is proved by 
mistakes or corruptions which they contain ; some others may be due 
to the copyist himself. 

V^ seldom has scholia common with the other older sources ; for 
the most part they either appear in V* alone or only in the later 
sources as v or P (later scholia in F), some being original, others not 
In Book x. V^ has three series of numerical examples, (i) with Greek 
numerals, (2) alternatives added later, also mostly with Greek numerals, 
(3) with Arabic numerals. The last class were probably the work of 
the copyist himself. As V^ belongs to the same time as the MS. 
(i2th c), these examples give an idea of the facility with which 
calculations were made in Byzantium at that time. They show too 
that the Greek method of writing numbers still preponderated in the 
nth c, but that the use of the Arabic numerals (in the East- Arabian 
form) was thoroughly established in the 12th c 

Of collections in other hands in V distinguished by Heiberg (see 
preface to Vol. v.), V* has very few scholia which are found in other 
sources, the greater part being original ; V*, V* are the work of the 
copyist himself; V^ are so in part only, and contain several scholia 
from Schol. Vat. and other sources. V* and V* are later than 13th 
— 14th c, since they are not found in f (cod. Laurent. XXVIII, 6) which 
was copied from V and contains, besides V* V^ the greater part of 
V^ and VI. No. 20 of V* (in the text). 

In P there are, besides P* (a quite late hand, probably one of the 
old Scriptores Graeci at the Vatican), two late hands (P*), one of 
which has some new and independent scholia, while the other has 




72 INTRODUCTION 

added the greater part of Schol Vm±, partly in the margin and 
partly on pieces of leaves stitched on, 

Our sources for Schol Vat also contain other elements. In P 
there were introduced a certain number of extracts from Proclus, to 
supplement Schol. Vat, to Book l. ; they are all written with a 
different ink from that used for the oldest part of the MS,, and the 
text is inferior. There are additions in the other sources of Schol 
Vat (F and B) which point to a common source for FB and which 
are nearly all found in other MSa, and, in particular, in Schol Vind., 
which also used the same source ; that they are not assignable to 
Schol Vat results only from their not being found in Vat Of other 
additions in F, some are peculiar to F and some common to it and b; 
but they are not original F* (scholia in a later hand in F) contains 
three original scholia ; the rest come from V. B contains, besides 
scholia common to it and F, b or other sources, several scholia which 
seem to have been put together by Arethas, who wrote at least a part 
of them with his own hand. 

Heiberg has satisfied himself, by a closer study of b, that the 
scholia which he denotes by b, fi and b* are by one hand ; they are 
mostly to be found in other sources as well though some are original 
By the same hand (Theodorus Cabasilas, 1 5th c.) are also the scholia 
denoted by W, B*, b* and B*. These scholia come in great part from 
Schol Vind., and in making these extracts Theodorus probably used 
one of our sources, 1, mistakes in which often correspond to those of 
Theodorus, To one scholium is attached the name of Demetrius (who 
must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c*); 
but it could not have been written by him, since it appears in B and 
Schol Vind. Nor are all the scholia which bear the name of 
Theodorus due to Theodorus himself, though some are so. 

As B* (a late hand in B) contains several of the original scholia of 
b", B* must have used b itself as his source, and, as all the scholia in 
B* are in b, the latter is also the source of the scholia in B* which are 
found in other MSS. B and b were therefore, in the 1 5th c, in the 
hands of the same person ; this explains, too, the fact that b in a late 
hand has some scholia which can only come from B, We arrive then 
at the conclusion that Theodorus Cabasilas, in the 15th c, owned both 
the MSS. B and b, and that he transferred to B scholia which he had 
before written in b, either independently or after other sources, and 
inversely transferred some scholia from B to b. Further, B' are 
earlier than Theodorus Cabasilas, who certainly himself wrote B* as 
well as b* and b*. 

An author's name is also attached to the scholia VI. No. 6 and 
X. No. 223, which are attributed to Maximus Planudes (end of 13th c) 
along with scholia on I. 31, x. 14 and x. 18 found in 1 in a quite late 
hand and published on pp. 46, 47 of Heiberg's dissertation. These 
seem to have been taken from lectures of Planudes on the Elements 
by a pupil who used 1 as his copy. 

There are also in 1 two other byzantine scholia, written by a late 
hand, and bearing the names loannes and Pediasimus respectively ; 






r 



t 



CH.VI] THE SCHOLIA 73 

these must in like manner have been written by a pupil after lectures 
of loannes Pediasimus (first half of 14th c), and this pupil must also 
have used 1. 

Before these scholia were edited by Heiberg, very few of them had 
been published in the original Greek. The Basel editio princeps has a 
few (v. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken, 
some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others 
probably from the Venice MS. (Marc. 301) also used by him; one 
published by Heiberg, not in his edition of Euclid but in his paper 
on the scholia, may also be from Venet. 301, but appears also in 
Paris. Gr. 2342. The scholia in the Basel edition passed into the 
Oxford edition in the text, and were also given by August in the 
* Appendix to his Vol. II. 

I Several specimens of the two series of scholia (Vat. and Vind.) 

' were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) 
I and by KnocYit {Untersuchungen fiber die neu aufgefundenen Scholien 
des Proklus, Herford, 1865). 

The scholia published in Latin were much more numerous. G. 
Valla {De expetettdis et fugiendis rebus^ 1501) reproduced apparently 
some 200 of the scholia included in Heiberg's edition. Several of 
these he obtained from two Modena MSS. which at one time were 
in his possession (Mutin. Ill B, 4 and II E, 9, both of the 15th c); 
but he must have used another source as well, containing extracts 
from other series of scholia, notably Schol. Vind. with which he has 
some 87 scholia in common. He has also several that are new. 

Commandinus included in his translation under the title *' Scholia 
antiqua" the greater part of the Schol. Vat. which he certainly 
obtained from a MS. of the class of Vat. 192; on the whole he 
adhered closely to the Greek text Besides these scholia Com- 
mandinus has the scholia and lemmas which he found in the Basel 
editio princeps^ and also three other scholia not belonging to Schol. 
Vat, as well as one new scholium (to xil. 13) not included in 
Heiberg's edition, which are distinguished by different type and were 
doubtless taken from the Greek MS. used by him along with the 
Basel edition. 

In Conrad Dasypodius' Lexicon mathematicum published in 1573 
there is (on fol. 42 — 44) "Graecum scholion in definitiones Euclidis 
libri quinti elementorum appendicis loco propter pagellas vacantes 
annexum." This contains four scholia, and part of two others, 
published in Heiberg's edition, with some variations of readings, and 
with some new matter added (for which see pp. 64 — 6 of Heiberg's 
pamphlet). The source of these scholia is revealed to us by another 
work of Dasypodius, Isaaci Monachi Scholia in Euclidis elententorum 
geometriae sex priores libros per C Dasypodium in latinutn sermonem 
translata et in lucent edita (1579). This work contains, besides 
excerpts from Proclus on Book I. (in part closely related to Schol. 
Vind.), some 30 scholia included in Heiberg's edition, several new 
scholia, and the above-mentioned scholia to the definitions of Book V. 
published in Greek in 1573. After the scholia follow ** Isaaci Monachi 




74 INTRODUCTION [c», vr 

prolegomena in Eudidis Elementorum geometriae Hbros" (two 
definitions of geometry) and "Van a miscellanea ad geometriae cogni- 
tionem necessana ab Isaaco Monacho collecta" (mostly the same as 
pp, 252^ 24 — 272, 27 in the Variae Colkcticnes included in Hultsch's 
Heron); lastly, a note of Dasypodius to the reader says that these 
schoha were taken '^ex clarissimi vrri Joan n is Sambuci anttquo cod ice 
many propria Isaaci Monachj scripto/* Isaak Monachus is doubtless 
Isaak Argyrus, 14th c; and Dasypodtus used a MS. in which, besides 
the passage in Hultsch's Variae CQikctwfus, were a number of 
scholia marked in the margin with the name of Isaak (c£ those in b 
under the name of Theodorus Cabasilas). Whether the new schoJia 
are original cannot be decided until they are published in Greek ; but 
it is not improbable that they are at all events independent arrange- 
ments of older scholia. All but five of the others, and all but one of 
the Greek scholia to Book v.^ are taken from Schol. Vat ; three of the 
excepted ones are from Schol, Vind,, and the other three seem to 
come from F (where some words of them are illegiblej but can be 
supplied by means of Mut in B, 4, which has these three scholia and 
generally shows a certain likeness to Isaak s scholia). 

Dasypodius also published in 1564 the arithmetical commentary 
of Barlaam the monk (14th c) on Eucl Book ll., which finds a place 
in Appendix IV, to the Scholia in Heiberg s edition. - 



CHAPTER VII. 



EUCLID IN ARABIA. 



We are told by Hajl Khalfa* that the Caliph al-Mansur (754-775) 
I sent a mission to the Byzantine Emperor as the result of which he 

(obtained from him a copy of Euclid among other Greek books, and 
again that the Caliph al-Ma'mun (813-833) obtained manuscripts of 
* Euclid, among others, from the Byzantines. The version of the 
} Elements by al-Hajjaj b. Yusuf b. Matar is, if not the very first, at 
least one of the first books translated from the Greek into Arabic*. 
According to the Fihrist^ it was translated by al-Hajjaj twice; the 
first translation was known as " Haruni " (" for Harun "), the second 
I bore the name "Ma'muni" ("for al-Ma'mun") and was the more trust- 
worthy. Six Books of the second of these versions survive in a Leiden 
MS. (Codex Leidensis 399, i) which is being published by Besthom 
and Heiberg*. In the preface to this MS. it is stated that, in the reign 
of Harun ar-RashId (786-809), al-Hajjaj was commanded by Yahya 
! b. Khalid b. Barmak to translate the book into Arabic Then, when 
^ al-Ma'mun became Caliph, as he was devoted to learning, al-Hajjaj 
saw that he would secure the favour of al-Ma'mun "if he illustrated and 
'. expounded this book and reduced it to smaller dimensions. He 
I' accordingly left out the superfluities, filled up the gaps, corrected or 
' removed the errors, until he had gone through the bcK)k and reduced 
it, when corrected and explained, to smaller dimensions, as in this 
I copy, but without altering the substance, for the use of men endowed 

• with ability and devoted to learning, the earlier edition, being left in 
'• the hands of readers." 

The FUifist goes on to say that the work was next translated by 
Ishaq b. Hunain, and that this translation was improved by Thabit b. 
Qurra. This Abu Ya'qub Ishaq b. Hunain b. Ishaq al-'ibadi (d. 910) 
was the son of the most famous of Arabic translators, Hunain b. Ishaq 
al-'ibadi (809-873), a Christian and physician to the Caliph al- 
Mutawakkil (847-861). There seems to be no doubt that Ishaq, who 

^ Lexicon bibliogr. et encychp, ed. FlUgel, III. pp. 91, 93. 

• Klamroth, Zeitschrift dtr Deutschen Morgenldndischen GeseUschafU XXXV. p. 303. 
' » Fihrist (tr. Sutcr), p. 16. 

^ Codtx Leidensis 399, i . Euclidis EUmenta ex interfretaiume al'Hadschdschadschii cum 

• ccmmeniariis al'NarizU^ Hauniae, part I. i. 1893, part I. ii. 1897, part II. i. 1905. 

i' 



r6 INTRODUCTION [ra. vu 

must have known Greek as well as hts father, made his translation 
direct from the Greek, The revision must apparently have been the 
subject of an arrangement between Ishaq and T ha bit, as the latter 
died in 90 ! or nine years before Ishaq, Thabit undoubtedly consulted 
Greek MSS. for the purposes of his revision. This is expressly stated 
in a marginal note to a Hebrew version of the Elements^ made from 
Ishaq's, attributed to one of two scholars belonging to the same family ^ 
viz, either to Moses b. Tibbon (about 1244-1274) or to Jakob b, Machir 
(who died soon after 1306)^ Moreover Thabit observes, on the pro- 
position which he gives as ix. 3 1, that he had not found this proposition 
and the one before it in the Greek but only in the Arabic ; from which 
statement Klamroth draws two conclusions, ( 1 ) that the Arabs had 
already begun to interest themselves in the authenticity of the text 
and (2) that Thabit did not alter the numbers of the propositions in 
Ishaq's translation'. The Fihrisi also says that Yuhanna al-Qass (i.e. 
" the Priest ") had seen in the Greek copy in his possession the pro- 
position in Book I. which Thabit took credit for, and that this was 
confirmed by Nazlf, the physician, to whom Yuhanna had shown it 
This proposition may have been wanting in Ishaq, and Thabit may 
have added it, but without claiming it as his own discovery'. As 
a fact, I. 45 is missing in the translation by al-Hajjaj, 

The original version of Ishaq wtthcut the improvements by ThSbit 
has probably not survived any more than the first of the two versions 
by al'Hajjaj ; the divergences between the MSS. are apparently due to 
the voluntary or involuntary changes of copyists, the former class 
varying according to the degree of mathematical knowledge possessed 
by the copyists and the extent to which they were influenced by 
considerations of practical utility for teaching purposes*. Two MSS* ' 
of the Ishaq -Thabit version exist in the Bodleian Library (No. 279 
belonging to the year 1238, and No. 280 written in i26o~i)"; Books 
L — XIIL are in the Ishaq-Thabit version, the non-Euclidean Books 
XIV,, XV. in the translation of Qusta b. Luqa al-Ba'labakkl (d. about 
912)* The first of these MSS. (No- 279) is that (O) used by Klamroth 
for the purpose of his paper on the Arabian Euclid. The other MS. 
used by Klamroth is (K) Kj^enhavn LXXXI, undated but probably 
of the 13th c, containing Books V. — XV., Books v. — X. being in the 
Ishaq-Thabit version, Books XL — XIII. purporting to be in al-Hajjaj's 
translation, and Books xiv., XV. in the version of Qusta b. Luqa. In 
not a few propositions K and O show not the slightest difference, and, 
even where the proofs show considerable diflferences, they are generally 
such that, by a careful comparison, it is possible to reconstruct the 
common archetype, so that it is fairly clear that we have in these cases, 
not two recensions of one translation, but arbitrarily altered and 

^ Steinfchneider, ZeiUchrift fiir Math, u, Pkysik, xxxi., hist-litt Abtheilung, pp. 85, 

"Klamroth, p. 379. * Stdmchndder, p. 88. 

^ Klamroth, p. 300. 

* These mss. are described by NicoU and Pusey, Caiaiogus cod, mss, orimi. diN. Bod- 
Umttoi, pt IL 1835 (pp. 357 — 262). 



I 



CH. vii] EUCLID IN ARABIA 77 

shortened copies of one and the same recension ^ <i^The Bodleian MS. 
No. 280 contains a preface, translated by Nicoll, which cannot be by 
Thabit himself because it mentions Avicenna (980-1037) and other 
later authors. The MS. was written at Mara^a in the year 1 260-1 and 
. has in the margin readings and emendations from the edition of 
f Naslraddin at-TusI (shortly to be mentioned) who was living at Maraga 
( at the time. Is it possible that at-TusI himself is the author of the 
I preface'? Be this as it may, the preface is interesting because it 
throws light on the liberties which the Arabians allowed themselves 
^ to take with the text. After the observation that the book (in spite 
r of the labours of many editors) is not free from errors, obscurities, 
L redundancies, omissions etc, and is without certain definitions neces- 
I sary for the proofs, it goes on to say that the man has not yet been 
' found who could make it perfect, and next proceeds to explain 
(i) that Avicenna *'cut out postulates and many definitions" and 
r attempted to clear up difficult and obscure passages, (2) that Abu'l 
I Wafa al-Buzjanl (939-997) "introduced unnecessary additions and 
I left out many things of great importance and entirely necessary," 
inasmuch as he was too long in various places in Book VI. and too 
short in Book X. where he left out entirely the proofs of the a^tamae, 
while he made an unsuccessful attempt to emend XII. 14, (3) that Abu 
Ja'far al-Khazin (d. between 961 and 971) arranged the postulates 
excellently but " disturbed the number and order of the propositions, 
reduced several propositions to one " etc. Next the preface describes 
the editor's own claims* and then ends with the sentences, " But we 
have kept to the order of the books and propositions in the woric itself 
(i.e. Euclid's) except in the twelfth and thirteenth books. For we have 
dealt in Book xiil. with the (solid) bodies and in Book Xll. with the 
surfaces by themselves." 

After Thabit the Fihrist mentions Abu 'Uthman ad-Dimashq! as 
having translated some Books of the Elements including Book X. (It 
is Abu 'Uthman's translation of Pappus* commentary on Book x. 
which Woepcke discovered at Paris.) The Fihrist adds also that 
" Nazlf the physician told me that he had seen the tenth Book of 
Euclid in Greek, that it had 40 propositions more than the version 
in common circulation which had 109 propositions, and that he had 
determined to translate it into Arabic." 

But the third form of the Arabian Euclid actually accessible to us 
is the edition of Abu Ja'far Muh. b. Muh. b. al-Hasan Naaraddin 
at-TusI (whom we shall call at-TusI for short), bom at Tus (in 
Khurasan) in 1201 (d. 1274). This edition appeared in two forms, a 
larger and a smaller. The larger is said to survive in Florence only 
(Pal. 272 and 313, the latter MS. containing only six Books) ; this was 
published at Rome in 1 594, and, remarkably enough, some copies of 

' Klamroth, pp. 306—8. 

* Steinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitichrift 
. , fiir Math, u. Physik, XXIX., hist.-Iitt. AUh. p. 16. 

* This seems to include a rearrangement of the contents of Books xiv., xv. added to the 
t Elements, 

> • 

V- 

^ ...^ 



* Cnrtse, 0^. ^. p. so; Heibeig, EukUd-Studim^ p. 178. 

• Hdberg^s Eudidy voL v. p. ci. 



*• Klamrotb, pp. 173— 4. 



ft INTRODUCTIOli^ [cb. ¥ii , 

this edition are to be found with 12 and some with 13 Books, some 
with a Latin title and some without\ But the book was printed tn 
Arabic, so that K^stner remarks that he will say as much about it as 
can be said about a book which one cannot read* The shorter form, 
which however, in most MSS., is in 15 Books, survives at Berlin, Munich, 
Oxford, British Museum (974, 1334*, 1335X P^ns (2465. 2466), India 
Office, and Constantinople ; it was printed at Constantinople in 
1801, and the first six Books at Calcutta in i824\ 

At*Tusl's work is however not a translation of Euclid's text, but a 
re*wr!tten Euclid based on the older Arabic translations. In this 
respect it seems to be like the Latin version of the EUmt^nts by 
Campanus {Cam pan o), which was first published by Erhard Ratdolt 
at Venice in J 482 (the first printed edition of Euclid*). Campanus 
(13th c.) was a mathematician^ and it is likely enough that he allowed 
himself the same liberty as at-TusI in reproducing Euclid. What- 
ever may be the relation between Campanus' version and that of 
Athelhard of Bath (about 1 120), and whether, as Curt^e thinks'^ they 
both used one and the same Latin version of loth — 1 Ith c, or whether 
Campanus used Athclhard's version in the same way as at-Tusi used 
those of his predecessors', it is certain that both versions came from 
an Arabian source, as is evident from the occurrence of Arabic words 
in them I Campanus' version is not of much service for the purpose 
of forming a judgment on the relative authenticity of the Greek and 
Arabian tradition ; but it sometimes preserves traces of the purer 
source, as when it omits Theons addition to wu 33'* A curious 
circumstance is that, while Campanus' version agrees with at-Tust's 
in the number of the propositions in all the genuine Euclidean Bcwks 
except V. and IX., it agrees with Athelhard's in having 34 propositions i 
in Book V. (as against 25 in other versions), which confirms the view ; 
that the two are not independent, and also leads, as Klamroth says, 
to this dilemma: either the additions to Book v. are Athelhard's 
own, or he used an Arabian Euclid which is not known to us^. 
Heiberg also notes that Campanus' Books xiv., xv. show a certain 
agreement with the preface to the Thabit-Ishaq version, in which the 
author claims to have (1) given a method of inscribing spheres in the 
five r^ular solids, (2) carried further the solution of the problem how 

' Sater, Die Mathematikir und Astronomm der Arader, p. 151. The Latin title is 
Eiulidis iUmeniorum geomeirkorum Wnri tredeeim. Ex traJitione doctissimi Nasiridim I 
Tktmi nunc frimnm arabice impressu Romae in typographia Medicea MDXCiv. Cum 
lioentia sapenonim. { 

* KMstner, Geschukie der Maihemaiik^ I. p. 367. I 

* Smer has a note that this MS. is very old, having been copied from the original in the y 
anthor's lifetime. ' 

* Suter, p. 151. 

* Described by Kistner, Geschickte der Mathemaiik^ i. pp. S89--399, and by Weiss- 
enbom, Die Ubersettungm da EukHd durch Campano und Zamberti^ Halle a. S., 188s, 
pp. I— -7. See also in/rat Chapter vni, p. 97. 

* Sonderabdruck des Jakresberichies iiSer die Fortschritte der kiassiuhen AUerthuwn- 
wissenschfai vom Oki. 1879—1881, Berlin, 1884. 

' Klamroth, p. 371. 



CH. vii] EUCLID IN ARABIA 79 

to inscribe any one of the solids in any other and (3) noted the cases 
where this could not be done\ 

With a view to arriving at what may be called a common measure 
of the Arabian tradition, it is necessary to compare, in the first place, 
the numbers of propositions in the various Books. Hajl Khalfa says 
that al-Hajjaj's translation contained 468 propositions, and Thabit's 
478 ; this is stated on the authority o( at-Tusi, whose own edition 
contained 468*. The fact that Thabit's version had 478 propositions 
is confirmed by an index in the Bodleian MS. 279 (called O by 
Klamroth). A register at the beginning of the Codex Leidensis 399, i 
which gives Ishaq's numbers (although the translation is that of 
al-Hajjaj) apparently makes the total 479 propositions (the number in 
Book XIV. being apparently 1 1, instead of the 10 of O*). I subjoin a 



1 


table ot relative nu 
the corresponding 


mbers tal 
numbers 


<en trom Kl 
in August's 


amroth, to which I have ad 
and Heiberg*s editions of 


1 


Greek text. 






- 






1 




The Arabian Euclid 


The Greek Euclid 




Books 


Ishaq 


at-TusI 


Campanus 


Gregory 


August 


Heiberg 


1 


I 


48 


48 


48 


48 


48 


48 


II 


14 


14 


14 


14 


14 


14 


• 


III 


36 


36 


36 


37 


37 


37 




IV 


16 


16 


16 


16 


16 


16 




V 


25 


25 


34 


25 


25 


25 


VI 


33 


32 


32 


33 


33 


33 




VII 


39 


39 


39 


41 


41 


39 




VIII 


27 


25 


25 


27 


. ^7 


27 




IX 


38 


36 


39 


36 


36 


36 




X 


109 


107 


107 


117 


116 


"5 




XI 


41 


41 


41 


40 


40 


39 




XII 


15 


IS 


15 


18 


18 


18 




XIII 


21 


18 


18 


18 


18 


18 






462 


452 


464 


470 


469 


465 




[xiv 


10 


10 


18 


7 




? 




XV 


6 


6 


13 


10 










478 


468 


495 


487 




?] 



The numbers in the case of Heiberg include all propositions which 
he has printed in the text; they include therefore XIII. 6 and III. 12 
now to be regarded as spurious, and X. 112 — 115 which he brackets 
as doubtful. He does not number the propositions in Books XIV., xv., 
but I conclude that the numbers in P reach at least 9 in XIV., and 9 
in XV. 

> Heiberg, Zeitichrift fur Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. i\, 

• Klamroth, p. 274; Steinschneider, Zeitichrift fiir Math, u, Physti, xxxi., hist.-litt. 
Abth. p. 98. 

* Besthom- Heiberg read ** 11 ?" asihe number, Klamroth had read it as 31 (p. 373). 



1 



80 INTRODUCTION [ch. to 

The Fihrist confirms the number 109 for Book X,, from which 
Klamroth concludes that Ishaq's version was considered as by far the 
most authoritative. 

In the text of O, Book iv, consists of 17 propositions and Book 
XIV, of 1 2, differing in this respect from its own table of contents ; IV. 
15, 16 in O are really two proofs of the same proposition. 

In al-Hajjaj's version Book L consists of 47 propositions only, 1, 45 
being omitted. It has also one proposition fewer in Book 11 1., the 
Heronic proposition ill. 12 being no doubt omitted* 

In speaking of particular propositions, I shall use Heiberg's 
numbering, except where otherwise stated. 

The difference of 10 propositions between Thabit*Ishaq and 
at-Tusi 15 accounted for thus : 

(i) The three propositions Vl 12 and x. 28, 29 which both Ishaq 
and the Greek text have are omitted in at-Tusi. 

(2) Ishaq divides each of the propositions Xlll. i — 3 into two, 
making six instead of three in at-Tus! and in the GreeL 

(3) Ishaq has four propositions (numbered by him viii. 24^ 25, 

IX. 30. 31) which are neither in the Greek Euclid nor in at*TusL 
Apart from the above differences aUHajjaj {so far as we know), 

Ishaq and at-Tusl agree \ but their Euclid shows many differences 
from our Greek text. These differences we will classify as follows ^ 

K Propositions. 

The Arabian Euclid omits VI L 20, 22 of Gregory's and August's 
editions (Heiberg, App. to Vol Ih pp. 428«32) ; VIIL 16, 17 ; X. 7, 8, 
13, 16, 24, 112, 113, 114, besides a lemma vtdgo X. 13, the proposition 

X. 117 of Gregory's edition, and the scholium at the end of the Book 
(see for these Heiberg*s Appendix to Vol. in, pp. 382, 408 — 416); 
XL 38 in Gregory- and August (Heibcrg, App. to Vol, Iv, p. 354); 
XII. 6, 13, 14; {also all but the first third of Book XV.). 

The Arabian Euclid makes II L 11, 12 into one proposition, and 
divides some propositions {X. 31, 32 ; XL 31, 34; xlll. i — 3) into two 
each. 

The order is also changed in the Arabic to the following extent 
V, 12, 13 are interchanged and the order in Books VI., VIL, IX. — 
xiiL is : 

VL 1—8, 13, II, 12. % 10, 14— *7» ift 20, 18, 21, 22| 24, 26, 23, 
25,27—30,32,31,33. 

VI I. I— 2Q, 22, 21, 23—28, 31, 32i 29, 30, 33 — 39. 

IX. 1-^13, 20, 14 — 19, 21 — 25, 27, 26, 28 — 36, with two new pro- 
positions coming before prop. 30. 

X. 1—6, 9—12, IS, 14, 17—23, 26—28, 25, 29—30, 31, 32, 33— 
111,115. 

XL I— 30, 3i> 32, 34, 33, 35— 39. 
XIL I— 5, 7, 9, 8, 10, 12, II, IS, 16—18. 
XIIL 1—3, s, 4, 6, 7, 12, 9, 10, 8, II, 13, IS, 14, 16—18. 

> See Klamroth, pp. 975—6, 38o» 383—4, 314— > 5* 3^6 ; Hdbeig, toL v. pp. xcvi, xcvii. 



k 



CH.VII] EUCLID IN ARABIA 8i 

2. Definitions, 

The Arabic omits the following definitions: iv. Deff. 3 — 7, vil. 
Def. 9 (or 10), XI. Deff. 5 — 7, 15, 17, 23, 25 — 28; but it has the 
spurious definitions VI. Deff. 2, 5, and those of proportion and ordered 
proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges 
V. Deff. II, 12 and also vi. Deff. 3, 4. 

The order of the definitions is also different in Book vil. where, 
after Def 11, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23, 
and in Book XI. where the order is i, 2, 3,4, 8. 10, 9, 13, 14, 16, 12, 21, 
22, 18, 19, 20, II, 24. 

3. Lemmas andporisms. 

All are omitted in the Arabic except the porisms to vi. 8, VIII. 2, 
X. 3 ; but there are slight additions here and there, not found in the 
Greek, e.g. in vill. 14. 15 (in K). 

4. Alternative proofs. 

These are all omitted in the Arabic, except that in X. 105, 106 they 
are substituted for the genuine proofs; but one or two alternative 
proofs are peculiar to the Arabic (Vl. 32 and vili. 4, 6). 

The analyses and syntheses to xill. i — 5 are also omitted in the 
Arabic. 

Klamroth is inclined, on a consideration of all these differences, to 
give preference to the Arabian tradition over the Greek (i) "on 
historical grounds," subject to the proviso that no Greek MS. as 
ancient as the 8th c. is found to contradict his conclusions, which are 
based generally (2) on the improbability that the Arabs would have 
omitted so much if they had found it in their Greek MSS., it being clear 
from the Fihrist that the Arabs had already shown an anxiety for a 
pure text, and that the old translators were subjected in this matter to 
the check of public criticism. Against the " historical grounds," Heiberg 
is able to bring a considerable amount of evidenced First of all there 
is the British Museum palimpsest (L) of the 7th or the b^inning of 
the 8th c. This has fragments of propositions in Book x. which are 
omitted in the Arabic; the numbering of one proposition, which agrees 
with the numbering in other Greek MS., is not comprehensible on 
the assumption that eight preceding propositions were omitted in it, 
as they are in the Arabic ; and lastly, the readings in L are tolerably 
like those of our MSS., and surprisingly like those of B. It is also to 
be noted that, although P dates from the loth c. only, it contains, 
according to all appearance, an ante-Theonine recension. 

Moreover there is positive evidence against certain omissions by 
the Arabians. At-Tusi omits VI. 12, but it is scarcely possible that, 
if Eutocius had not had it, he would have quoted VI. 23 by that 
number*. This quotation of VI. 23 by Eutocius also tells against 
Ishaq who has the proposition as VI. 25. Again, Simplicius quotes VI. 
10 by that number, whereas it is VI. 13 in Ishaq ; and Pappus quotes, 
by number, XIII. 2 (Ishaq 3, 4), XIII. 4 (Ishaq 8), XlII. 16 (Ishaq 19). 

1 Heiberg in Zeitschrift fur Math, u, PhysiJk, XXIX., hist.-litt Abth. p. asqq. 
' Apollonius, ed. Heibeig, vol. 11. p. 318, 3 — 5. 

H. E. 6 



it INTRODUCTION 

On the other hand the contraction of IIL n, 12 into one propositton 
in the Arabic tells in favour of the Arabic. 

Further, the omission of certain porisms in the Arabic cannot be 
supported; for Pappus quotes the porism to Xlll. 17*, P rod us those 
to I [. 4, III, I, VI L 2\ and Simplicius that to iv, 15. 

Lastly, some propositions omitted in the Arabic are required in 
later propositions. Thus X. 13 is used in X, 18, 22, 23, 26 etc. ; X. 17 
is wanted in x, 18, 26^ 36; Xlh 6, 13 are required for XIL 11 and XIL 
15 respectively. 

It must also be remembered that some of the things which were 
properly omitted by the Arabians are omitted or marked as doubtful 
in Greek MSS, also, especially in P, and others are rightly suspected for 
other reasons (ag. a number of alternative proofs, lenr^mas, and porisms, 
as well as the analyses and syntheses of Xlil. i — 5). On the other 
hand, the Arabic has certain interpolations peculiar to our inferior 
MSS, (cf the definition VL Def 2 and those of fitafiorlian and ordered 
proportion). 

Heiberg comes to the general conclusion that, not only is the 
Arabic tradition not to be preferred offhand to that of the Greek MSs., 
but it must be regarded as inferior in authority. It is a question 
how far the differences shown in the Arabic are due to the use of 
Greek MSS, differing from those which have been most used as the 
basis of our text, and how far to the arbitrary changes made by 
the Arabians themselves. Changes of order and arbitrary omissions 
could not surprise us, in view of the preface above quoted from the 
Oxford MS, of Thabit-Ishaq, with its allusion to the many important 
and necessary things left out by Abu *1 Wafa and to the author's 
own rearrangement of Books XIL, XIII. But there is evidence of 
differences due to the use by the Arabs of other Greek MSS. Heiberg' 
is able to show considerable resemblances between the Arabic text 
and the Bologna MS, b in that part of the MS. where it diverges so 
remarkably from our other MSS. (see the short description of it above, 
p, 49); in illustration he gives a comparison of the proofs of xtt, 7 in b 
and in the Arabic respectively, and points to the omission in both of 
the proposition given in Gregozy's edition as XI, 3S, and to a remark- 
able agreement t>etween them as r^ards the order of the propositions 
of Book XIL As above stated, the remarkable divergence of b only 
affects Books xi. (at end) and xii. ; and Book xiii. in b shows none 
of the transpositions and other peculiarities of the Arabic There 
are many differences between b and the Arabic^ especially in the 
definitions of Book XI., as well as in Book XIII. It is therefore a 
question whether the Arabians made arbitrary changes, or the Arabic 
form is the more ancient, and b has been altered through contact 
with other MSS. Heiberg points out that the Arabians must be sdone 
responsible for tlieir definition of a prism, which only covers a prism 
with a triangular base. This could not have been Euclid's own, for 
the word prism already has the wider meaning in Archimedes, and 

* Pappus, V. p. 436, 5. • Prodiu, pp. 303 — ^4. 

* ZeUsckr^fiir Maih.m. Pkysik^ XXIX., bist.-litt. Abth. p. 6aqq. 



, 



i 



IF 



' 






CH. vii] EUCLID IN ARABIA 83 

Euclid himself speaks of prisms with parallelograms and polygons 
as bases (xi. 39 ; XII. 10). Moreover, a Greek would not have been 
likely to leave out the definitions of the " Platonic " regular solids. 

Heiberg considers that the Arabian translator had before him 
a MS. which was related to b, but diverged still further from the rest 
of our MSS. He does not think that there is evidence of the existence of 
a redaction of Books I. — x. similar to that of Books XL, XII. in b ; for 
Klamroth observes that it is the Books on solid geometry (xi. — XIII.) 
which are more remarkable than the others for omissions and shorter 
proofs, and it is a noteworthy coincidence that it is just in these 
Books that we have a divergent text in b. 

An advantage in the Arabic version is the omission of VII. DeC 10, 
although, as lamblichus had it, it may have been deliberately omitted 
by the Arabic translator. Another advantage is the omission of the 
analyses and syntheses of xill. i — 5 ; but again these may have been 
omitted purposely, as were evidently a number of porisms which 
are really necessary. 

One or two remarks may be added about the Arabic versions 
as compared with one another. Al-Hajjaj*s object seems to have 
been less to give a faithful reflection of the original than to write 
a useful and convenient mathematical text-book. One characteristic 
of it is the careful references to earlier propositions when their results 
are used. ' Such specific quotations of earlier propositions are rare in 
Euclid ; but in al-Hajjaj we find not only such phrases as "by prop, 
so and so," " which was proved " or " which we showed how to do in 
prop, so and so," but also still longer phrases. Sometimes he repeats 
a construction, as in 1. 44 where, instead of constructing "the parallelo- 
gram BEFG equal to the triangle C in the angle EBG which is equal 
to the angle Z>" and placing it in a certain position, he produces AB 
to G, making BG equal to half DE (the base of the triangle CDE in 
his figure), and on GB so constructs the parallelogram BHKG by 
I. 42 that it is equal to the triangle CDE, and its angle GBH is equal 
to the given angle. 

Secondly, al-Hajjaj, in the arithmetical books, in the theory of 
proportion, in the applications of the Pythagorean 1. 47, and generally 
where possible, illustrates the proofs by numerical examples. It is 
true, observes Klamroth, that these examples are not apparently 
separated from the commentary of an-NairIzi, and might not there- 
fore have been due to al-Hajjaj himself; but the marginal notes to 
the Hebrew translation in Munich MS. 36 show that these additions 
were in the copy of al-Hajjaj used by the translator, for they expressly 
give these proofs in numbers as variants taken from al-Hajjaj ^ 

These characteristics, together with al-Hajjaj*s freer formulation 
of the propositions and expansion of the proofs, constitute an in- 
telligible reason why Ishaq should have undertaken a fresh translation 
from the Greek. Klamroth calls Ishaq's version a model of a good 
translation of a mathematical text ; the introductory and transitional 

1 Kkmroth, p. 310; Steinschneider, pp. 85 — 6. 

6 — 2 



84 INTRODUCTION *^ [ch. vu 

phrases are stereotyped and few in number^ the technical terms are 
simply and consistently rendered, and the less formal expressions 
connect themselves as closely with the Greek as is consistent with 
intelligibility and the character of the Arabic language. Only in 
isolat^ cases does the formulation of definitions and enunciations 
differ to any considerable extent from the original. In general, his 
object seems to have been to get rid of difficulties and unevennesses 
in the Greek text by neat devices, while at the same time giving a 
faithful reproduction of it^ 

There are curious points of contact between the versions of 
al-Iiajjaj and Thabit-Ishaq. For example, the definitions and 
enunciations of propositions are often word for word the same» 
Presumably this is owing to the fact that Ishaq found these de- 
finitions and enunciations already established in the schools in his 
time, where they would no doubt be learnt by heart, and refrained 
from translating them afresh, merely adopting the older version with 
some changes*. Secondly, there is remarkable agreement between 
the Arabic versions as regards the figures, which show considerable 
variations from the figures of the Greek text, especially as regards 
the letters ; this is also probably to be explained in the same way, 
all the later translators having most likely borrowed al^Hajjaj's 
adaptation of the Greek figures'. Lastly, it is remarkable that the 
version of Books XL — xni, in the Kjjrfbenhavn Ms. (K), purporting 
to be by al-Hajjaj, is almost exactly the same as the Thabtt- Ishaq 
version of the same Books in O. Klamroth conjectures that Ishaq 
may not have translated the Books on solid geometry at all, and that 
Thabit took them from al-yajjaj, only making some changes in order 
to fit them to the translation of Ishaq*. 

From the facts (i) that at-Tusi's edition had the same number 
of propositions (468) as al-Hajjaj's version, while Thabit- Ishaq's had 
478, and (2) that at-Tusi has the same careful references to earlier 
propositions, Klamroth concludes that at-TusI deliberately preferred 
al-^ajjaj's version to that of Ishaq*. Heiberg, however, points out 
(i) that at-TusI left out Vl iz which, if we may judge by Klamroth's 
silence, al-I;^ajjaj had, and (2) al-Hajjaj's version had one proposition 
less in Books L and IlL than at-Tusi has. Besides^ in a passage quoted 
by I:Iaji Khalfa* from at-TusI, the latter says that ** he separated the | 
things which, in the approved editions, were taken from the archetype j 
from the things which had been added thereto," indicating that he ; 
had compiled his edition from dotA the earlier translations'. i 

There were a large number of Arabian commentaries on, or ; 
reproductions of, the Elements or portions thereof, which will be 1 

1 Klamroth, p. 390, illnstrates Is^*s method hy hit way of distiiigaifhin^ i^aptnb^ i 

(to be oongnient with) and 4^apiUfwBai fto be applied to), the confusion of which by timnt- J 

laton was animadverted on by Savile. uStjIkxi avoided the confusion by using two entirely < 

different words. ^ 

' Klamroth, pp. 310 — i. * ibid, p. aSy. ' 

^ iHd, pp. 304 — 5. * Uid. p. 374. 



• tMd, pp. 304—5. 

• miKhia&,i. p. 383. 
' Hoberg, Zeitsckrififir 



Math. m. Pkysik^ xxix., hist-litt. Abth. pp. a, 3. 



CH. vii] EUCLID IN ARABIA 85 

found fully noticed by Steinschneider^ I shall mention here the 
commentators etc. referred to in the Fihrist, with a few others. 

1. Abu '1 'Abbas al-Fadl b. Hatim an-NairlzI (bom at Nairiz, 
died about 922) has already been mentioned*. His commentary 
survives, as r^^rds Books I. — vi., in the Codex Leidensis 399, i, now in 
course of publication by Besthorn and Heiberg, and as regards 
Books I. — ^X. in the Latin translation made by Gherard of Cremona 
in the nth c. and now published by Curtze from a Cracow MS.' Its 
importance lies mainly in the quotations from Heron and Simplicius. 

2. Ahmad b. 'Umar al-Kar&blsI (date uncertain, probably 9th — 
loth c), ''who was among the most distinguished geometers and 
arithmeticians*." 

3. A1-' Abbas b. Sa'ld al-Jauharl (fl. 830) was one of the astro- 
nomical observers under al-Ma*mun, but devoted himself mostly to 
geometry. He wrote a commentary to the whole of the Elements^ 
from the beginning to the end ; also the " Book of the propositions 
which he added to the first book of Euclid*." 

4. Muh. b. 'Isa Abu 'Abdallah al-Mahtal (d. between 874 and 
884) wrote, according to the Fihrist, (i) a commentary on Eucl. 
Book v., (2) " On proportion," (3) " On the 26 propositions of the 
first Book of Euclid which are proved without reductio ad absurdum^' 
The work " On proportion " survives and is probably identical with, or 
part of, the commentary on Book v.' He also wrote, what is not 
mentioned by the Fihrht^ a commentary on Eucl. Book X., a fragment 
of which survives in a Paris MS.* 

5. Abu Ja'far al-Khftzin (i.e. " the treasurer " or '* librarian "), one 
of the first mathematicians and astronomers of his time, was born in 
Khurasan and died between the years 961 and 971. The Fikrist 
speaks of him as having written a commentary on the whole of the 
Elements*^ but only the commentary on the beginning of Book x. 
survives (in Leiden, Berlin and Paris) ; therefore either the notes on 
the rest of the Books have perished, or the Fikrist is in error *•. The 
latter would seem more probable, for, at the end of his commentary, 
al-Khazin remarks that the rest had already been commented on by 
Sulaiman b. 'Usma (Leiden MS.)" or 'Oqba (Suter), to be mentioned 
below. Al-Khazin's method is criticised unfavourably in the preface 
to the Oxford MS. quoted by Nicoll (see p. y7 above). 

6. Abii '1 Wafa al-BuzjanI (940-997), one of the greatest 
Arabian mathematicians, wrote a commentary on the Elements, but 

^ Steinschneider, Zeitschriftfur Math, u, Physik^ xxxi., hist.-litt. Abth. pp. 86 sqq. 
' Steinschneider, p. 86, Pihrist (tr. Suter), pp. 16, 67 ; Suter, Die Mathematiier und 
AHronomen tier Araber (1900), p. 45. 

' Supplementum ad Eucltdis opera omnia^ ed. Heiberg and Menge, Leipzig, 1899. 

* Fihrist^ pp. 16, 38 ; Steinschneider, p. 87 ; Suter, p. 65. 

* Fihrisi^ pp. 16, 35; Steinschneider, p. 88; Suter, p. n. 

* Fikrist, pp. 16, 25, 58. 

' Suter, p. 36, note, quotes the Paris MS. 3467, 16** containing the work "on proportion'* 
as the authority for this conjecture. 

* MS. 3457, 39<» (cf. Woepcke in Mhn, pris, h Facad. des sciinces, xiv., 1856, p. 669). 

* Fikrist, p. 17. *• Suter, p. 58, note b. " Steinschneider, p. 89. 



86 INTKODUCnOH 



^micu* Vtt I ^! 



did not complete it*. His method is also unfavourably iri^rded in I 
the same preface to the Oxford MS. 28a According to Haji Khalfa, he 
also wrote a book on geometrical constructions, in thirteen chapters. 
Apparently a book answering to this description was compiled by a 
gifted pupil from lectures by Abu 1 Wafa, and a Paris MS. (Anc fends 
169) contains a Persian translation of this work^ not that of Abu '1 Wafa 
himself. An analysis of the work was given by Wocpcke', and some 
particulars will be found in Cantor'* Abu '1 Wafa also wrote a 
commentary on Diophantus, as well as a separate " book of proofs 
to the propositions which Diophantus used in his book and to what 
he (Abu '1 Wafa) employed in his commentary*/' 

7. Ibn Rfthawaihl al-ArjinT also commented on EucI, Book X,* 

8. 'All b. Ahmad Abu U-Qasim al-AntikI (d. 987) wrote a 
commentary on the whole book*; part of it seems to survive (from 
the 5th Book onwards) at Oxford (CataL MSS. orient. II. 281 )^ 

9. Sind b. *Ali Abu *t-Taiyib was a Jew who went over to 
Islam in the time of aKMa'niun, and was received among his astro- 
nomical observers, whose head he became" (about 830) ; he died after 
864. He wrote a commentary on the whole of the EkmtHts ; " Abu 
*AlI saw nine books of it^ and a part of the tenthV His book " On 
the Apotomae and the Medials " mentioned by the Fikrisi^ may be 
the same as, or part of, his commentary on Book X* 

10. Abu Yusuf Ya'qub b. Muh. ar-Razf '^ wrote a commentafy 
on Book X., and that an excellent one, at the instance of Ibn al- 
*AmId"." 

11. The Fihrisi next mentions al> Kindt (Abu Yusuf Ya'qQb b. 
Ishaq b. as-Sabbah al-Kindi, d. about 873), as the author (1) of a 
work " on the objects of Euclid's book/' in which occurs the statement 
that the Elements were originally written by Apollonius, the carpenter 
(see above, p. 5 and note), (2) of a book ''on the improvement of 
Euclid's work," and (3) of another " on the improvement of the 14th 
and 15th Books of Euclid." ''He was the most distinguished man 
of his time, and stood alone in the knowledge of the old sciences 
collectively ; he was called ' the philosopher of the Arabians ' ; his 
writings treat of the most different branches of knowledge, as logic, 
philosophy, geometry, calculation, arithmetic, music, astronomy and 
others"." Among the other geometrical works of al-Kindi mentioned \ 
by the Fihrisi^ are treatises on the closer investigation of the results ! 
of Archimedes concerning the measure of the diameter of a circle in ^' 
terms of its circumference, on the construction of the figure of the two \ 
mean proportionals, on the approximate determination of the chords 1 

• Fikristt p. 17. I 

• WoepdLcyimmaiAnaH^tte, Sit. v. T. v. pp. «i8— 956 and 300—350. l 

• Geuk. d. Math, vol. ig, pp. 743—6. 
« mhriti^ p. 39 ; Suter, p. 71. • Rkrisi, p. 17 ; Sater, p. 17. 

• Rkrisi^ p. 17. f Suter, p. 64. 

• Fihrisi^ p. 17, 39 ; Suter, pp. 13, 14. • Hhri$t, p. 17. 
M Fihrist^ p. 17 ; Suter, p. 66. » FihrisU p. 17, ia-15. 
^ The mere catalogue of al-Kindl't works on the various boAclies of idence takes up 

Ibpr octavo pages (i i— 15) of Suter's translation of the Ftkrist. 



1 



CH. vii] EUCLID IN ARABIA 87 

of the circle, on the approximate determination of the chord (side) of 
the nonagon, on the division of triangles and quadrilaterals and con- 
structions for that purpose, on the manner of construction of a circle 
which is equal to the surface of a given cylinder, on the division of 
the circle, in three chapters etc. 

12. The physician Nazlf b. Yumn (or Yaman) al-Qass ("the 
priest") is mentioned by the Fihrist as having seen a Greek copy 
of Eucl. Book X. which had 40 more propositions than that which 
was in general circulation (containing 109), and having determined 
to translate it into Arabic^ Fragments of such a translation exist 
at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab, in 
Woepcke's tract); No. 18 contains "additions to some propositions 
of the loth Book, existing in the Greek language"." Nazlf must have 
died about 990'. 

13. Yu^iannft b. Yusuf b. al-Iiarith b. al-Bitnq al-Qass (d. about 
980) lectured on the Elements and other geometrical books, made 
translations from the Greek, and wrote a tract on the " proof" of the 
case of two straight lines both meeting a third and making with it, 
on one side, two angles together less than two right angles*. Nothing 
of his appears to survive, except that a tract "on rational and irrational 

^ magnitudes," No. 48 in the Paris MS. just mentioned, is attributed 
' to hinL 

I 14. Abu Muh. al-Hasan b. 'Ubaidallah b. Sulaiman b. Wahb 

. (d. 901) was a geometer of distinction, who wrote works under the 
y two distinct titles " A commentary on the difficult parts of the work 
r of Euclid " and " The Book on Proportion*." Suter thinks that an- 
other reading is possible in the case of the second title, and that it 
may refer to the Euclidean work "on the divisions (of figures)*." 
, 15. Qustft b. Luq§ al-Ba'labakki (d. about 912), a physician, 

philosopher, astronomer, mathematician and translator, wrote " on the 
difficult passages of Euclid's book" and "on the solution of arith- 
metical problems from the third book of Euclid'"; also an "intro- 
duction to geometry," in the form of question and answer". 

16. Thabit b. Qurra (826-901), besides translating some parts 
of Archimedes and Books V. — vii. of the Conies of Apollonius, and 
revising Ishaq*s translation of Euclid's Elements^ also revised the trans- 
lation of the Data by the same Ishaq and the book On divisions of 
figures translated by an anonymous writer. We are told also 
that he wrote the following works: (i) On the Premisses (Axioms, 
Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On 
the propositions and questions which arise when two straight lines 
are cut by a third (or on the " proof" of Euclid's famous postulate). 
The last tract is extant in the MS. discovered by Woepcke (Paris 
2457, 32®). He is also credited with "an excellent work" in the 
shape of an " Introduction to the Book of Euclid," a treatise on 



* Fikristf pp. 16, 17. 

" Woepcke, Mhn.pris. h Vacad, des sdenut^ XI v. pp. 666, 668. 

» Suter, p. 68. * Fihnst, p. 38 ; Suter, p. 

• Fihrist^ p. «6, and Suter's note, p. 60. • Suter, p. an, note 13. 

' Fihrist, p. 43. • Fihrist^ p. 43 ; Suter, p. 



60. 
41. 



wm 



88 INTRODUCTION [ch. vii 



I 



Geometry dedicated to fsma^n b. Bulbul, a Compendium of Geometry, 
and a large number of other works for the titles of which reference 
may be made to Sutcr, who also gives particulars as to which are 
extant^ 

17. Abu Sa'ld Sinin b* Thabit b, Qurra, the son of the translator 
of Euclid, followed in his father's footsteps as geometer, astronomer 
and physician. He wrote an ''improvement of the book of ,.-,.. on 
the Elements of Geometry, in which he made various additions to the 
original" It is natural to conjecture that Emlid is the name missing 
in 3iis description (by Ibn abl Usaibi'a); Casiri has the name Aqaton*. 
The latest eaitor of the Tdrlkh al-Hukantd^ however^ makes the name 
to be Iflaton (« Plato), and he refers to the statement by the Fihrist 
and Ibn" al-QiftI attributing to Plato a work on the Elements of 
Geometry translated by Qusta. It is just possible, therefore, that at 
the time of Qusta the Arabs were acquainted with a book on the 
Elements of Geometry translated from the Greek, which they attri- 
buted to Plato*. Sinan died in 943. 

18. Abu Sahl Wijan (or Waijan) b. Rustam al-Kuhl (fl, 988), 
bom at Kuh in Tabaristan, a distinguished geometer and astronomer, 
wrote, according to the Fihrist, a " Book of the Elements " after that 
of Euclid^; the ist and 2nd Books survive at Cairo, and a part of 
the 3rd Book at Berlin {^<^22Y, He wrote also a number of other 
geometrical works: Additions to the 2nd Book of Archimedes on . 
the Sphere and Cylinder (extant at Paris, at Leiden* and in the India ( 
Office), On the finding of the side of a heptagon in a circle (India 
Office and Cairo), On two mean proportionals (India OfficeX which 
last may be only a part of the Additions to Archimedes' On the Sphere -% 
and Cylinder, etc '\ 

19. Abu Nasr Muh. b. Muh. b. Tarkhan b. Uzlag al-FtrftbT 
(870-950) wrote a commentary on the difficulties of the introductory 
matter to Books I. and v.* This appears to survive in the Hebrew 
translation which is, with probability, attributed to Moses b. Tibbon^ i 

2a Abu 'All al-^asan b. al-Hasan b. al-Haitham (about 965- ] 
1039), known by the name Ibn al-Haitham or Abu 'All al-Basrl, was a 
man of great powers and knowledge, and no one of his time approached I 
him in the field of mathematical science. He wrote several works on 
Euclid the titles of which, as translated by Woepcke from Usaibi'a, i 
are as follows" 

1. Commentary and abridgment of the Elements. i 

2. Collection of the Elements of Geometry and Arithmetic, f 
drawn from the treatises of Euclid and ApoUonius. j 

3. Collection of the Elements of the Calculus deduced from 
the principles laid down by Euclid in his Elements. \ 

* Sater, pp. 54—8. 

* Purist (ed. Suter), p. 59, note 13a ; Suter, p. 51, note b. 

■ See Suter in Bibhatkeca Matkematica, iv„ 1903-4, pp. 196—7, reriew of Julias 
Lippert's Km ai-Qifii. Ta'ruh al-kukamd, Leiprig, 1903. 

* Fikristt p. 40. • Suter, p. 75. 

* Suter, p. 55. f SteioKhneider, p. 9«. 

* Stcintcnncider, pp. 93 — ^3. 



I 



r 



i 



1 



CH. vii] EUCLID IN ARABIA 89 

4. Treatise on "measure" after the manner of Euclid's 
Elements. 

5. Memoir on the solution of the difficulties in Book i. 

6. Memoir for the solution of a doubt about Euclid, relative 
to Book V. 

7. Memoir on the solution of a doubt about the stereometric 
portion. 

8. Memoir on the solution of a doubt about Book Xil. 

9. Memoir on the division of the two magnitudes mentioned 
in X. I (the theorem of exhaustion). 

ID. Commentary on the definitions in the work of Euclid 
(where Steinschneider thinks that some more general expression 
should be substituted for " definitions "). 

The last-named work (which Suter calls a commentary on the 
Postulates of Euclid) survives in an Oxford MS. (CataL MSS. orient. 
I. 908) and in Algiers (1446, i®). 

A Leiden MS. (966) contains his Commentary "on the difficult 
places " up to Book v. We do not know whether in this commentary, 
which the author intended to form, with the commentary on the 
Musadarat, a sort of complete commentary, he had collected the 
separate memoirs on certain doubts and difficult passages mentioned 
in the above list. 

A commentary on Book v. and following Books found in a 
Bodleian MS. (Catal. II. p. 262) with the title " Commentary on Euclid 
and solution of his difficulties " is attributed to b. Haitham ; this might 
be a continuation of the Leiden MS. 

The memoir on x. i appears to survive at St Petersburg, Ma de 
rinstitut des langues orient 192, 5* (Rosen, Catal. p. 125). 

21. Ibn SInft, known as Avicenna (980-1037), wrote a Com- 
pendium of Euclid, preserved in a Leiden MS. No. 1445, and forming 
the geometrical portion of an encyclopaedic work embracing Logic, 
Mathematics, Physics and Metaphysics^ 

22. Ahmad b. al-Husain al-AhwftzT al-Katib wrote a com- 
mentary on Book X., a fragment of which (some 10 pages) is to be 
found at Leiden (970), Berlin (5923) and Paris (2467, i8'*)*. 

23. Naslraddin at-Tusi (i 201-1274) who, as we have seen, 
brought out a Euclid in two forms, wrote: 

1. A treatise on the postulates of Euclid (Paris, 2467, 5®). 

2. A treatise on the 5th postulate, perhaps only a part of 
the foregoing (Berlin, 5942, Paris, 2467, 6®). 

3. Principles of Geometry taken from Euclid, perhaps 
identical with No. i above (Florence, Pal. 298). 

4. 105 problems out of the Elements (Cairo). He also edited 
the Data (Berlin, Florence, Oxford etc.)'. 

24. Muh. b. Ashraf Shamsaddin as-Samarqandl (fl. 1276) wrote 
" Fundamental Propositions, being elucidations of 35 selected proposi- 

^ Steinschneider, p. 91 ; Suter, p. 89. ' Suter, p. 57. 

• Suter, pp. 150 — I. 



u 



^a 



90 INTRODUCTION [cm. vii 

tions of the first Books of Euclid " which are extant at Gotha (1496 
and 1497), Oxford (Catal l. 967, 2^), ^^^ Brit Uus.\ 

25, Musa b- Muh. b* Mahmud, known as Qa^izade ar-Ruml (i,c» 
the son of the judge from Asia Minor), who died between 1436 and 
1446, wrote a commentary on the "Fundamental Propositions" ju^t 
mentioned^ of which many MSS* are extant'. It contained biographical 
statements about Euclid alluded to above (p. S, note), 

26, Abu Da'ud Sulaiman b, 'Uqba^ a contemporary of aKKhazin 
(see above, No* 5), wrote a commentary on the second half of Book x., 
which is, at least partly, extant at Leiden (974) under the title " On 
the binomials and apotomae found in the lOth Book of Euclid V 

27, The Codex Letdensis 399^ i containing al-Hajjaj*s transla- 
tion of Books L — VL is said to contain glosses to it by Sa'id b* Mas'ud 
b. al-Qass, apparently identical with Abu Nasr Gars al-Na'ma, son of 
the physician Mas'^Od b. al-Qass al-Bagdadi, who lived in the time of 
the last Caliph al-Musta'sini (d. 1258)*, 

28, Abu Muhammad b. Abdalbiql al-Bagdadt al*Fara^i (d* 
1 141, at the age of over 70 years) is stated in the Tdrtkh al-Hukamd 
to have written an excellent commentary on Book x. of the Elements, 
in which he gave numerical examples of the propositionsV This is 
published in Curtze's edition of an-Nainzi where it occupies pages 
252 — 386* 

29, Yahya b, Muh. b, 'Abdan b. *Abdalwahid, known by the 
name of Ibn al-Lubudl (1210-126S), wrote a Compendium of Euclid^ 
and a short presentation of the postulates'. 

30, Abu 'Abdallah Muh. b, Mu'adh al-Jayyanl wrote a com- 
mentary on Eucl, Book v. which survives at Algiers (1446, 3*)'. 

31, Abu Nasr Mansur b* 'All b. 'Iraq wrote^ at the instance of 
Muh. b, Ahmad Abu V-Raihan al-Blruni (973-1048), a tract "on 
a doubtful (difficult) passage in Eucl Book XIII/' (Berlin, S925)'. 



' Stit«T, p. 157- * *^*^' P" *75- * *^" P- 5**' 

< ihid. pp, [55— 4 i *V' 

° GaiU, i^ i4 ; Suloschiieider, pp. 94 — 5. 

* Safer in BWioiMsca MaiAemaiua^ iv^, 1904, pp. «5» 195 ; Sater has also an artide on 
its contents, BiUioikica MaiJkimaiuat viu, 1906^71 pp. 354 — 351. 

^ Steinschneider, p. 94 ; Sater, p. 14O. 

> Sater, Nacktrdge und Berickttgwtgen^ in Abkandimngen mr Gesck. der math, fVUsin- 
sehapen^ xiv., 1903, p. 17a 

* Snter, p. 81, ana Na€htrag$^ p. 173. 



CHAPTER VIII. 

PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS. 

Cicero is the first Latin author to mention Euclid^; but it is not 
likely that in Cicero's time Euclid had been translated into Latin or 
was studied to any considerable extent by the Romans; for, as Cicero 
says in another place', while geometry was held in high honour 
among the Greeks, so that nothing was more brilliant than their 
mathematicians, the Romans limited its scope by having regard only 
to its utility for measurements and calculations. How very little 
theoretical geometry satisfied the Roman agrimensares is evidenced 
by the work of Balbus de mensuris^^ where some of the definitions of 
Eucl. Book I. are given. Again, the extracts from the Elements found 

. in the fragment attributed to Censorinus (fl. 238 A.D )* are confined to 

I the definitions, postulates, and common notions. But by degrees the 
Elements passed even among the Romans into the curriculum of a 

! liberal education ; for Martianus Capella speaks of the effect of the 
enunciation of the proposition "how to construct an equilateral 
triangle on a given straight line " among a company of philosophers, 

' who, recognising the first proposition of the Elements, straightway 
break out into encomiums on Euclid*. But the Elements were then 

; {c. 470 A.D.) doubtless read in Greek ; for what Martianus Capella 
gives* was drawn from a Greek source, as is shown by the occurrence 
of Greek words and by the wrong translation of I. def. i (" punctum 
vero est cuius pars nihil est"). Martianus may, it is true, have 
quoted, not from Euclid himself, but from Heron or some other ancient 
source. 

But it is clear from a certain palimpsest at Verona that some 
scholar had already attempted to translate the Elements into Latin. 
This palimpsest' has part of the " Moral reflections on the Book of 
Job " by Pope Gregory the Great written in a hand of the 9th c. above 
certain fragments which in the opinion of the best judges date from 
the 4th c Among these are fragments of Vergil and of Livy, as well 
as a geometrical fragment which purports to be taken from the 14th 
and 15th Books of Euclid. As a matter of fact it is from Books XIL 
and XIII. and is of the nature of a free rendering, or rather a new 

* Dearatort III. 13a. • Tusc, I. 5. 

* GromaHH veteres, I. 97 sq. (ed. F. Blume, K. Lachmann and A. Rudorff. Berlin, 
1848, 1851). 

* Censorinus, ed. Hultsch, pp. 60— 3. 

* Martianus Capella, vi. 734. * ibid, vi. 708 sq. 
^ Cf. Cantor, i„ p. 565. 



:; 



INTRODUCTION [ch, viii 



arrangement, of Euclid with the propositions in different orders The 
MS- was evidently the translator's own copy, because some words arc 
struck out and replaced by synonyms. We do not know whether the 
translator completed the translation of the whole* or in what relation 
his version stood to our other sources. 

Magnus Aurelius Cassiodorius (b* about 475 a.d.) in the geometrical 
part of his encyclopaedia Deartibus ac disciplinis Hberalium literarum 
says that geometry was represented among the Greeks by Euclid, 
Apollonius, Archimedes, and others, "of whom Euclid was given us 
translated into the Latin language by the same great man Boethius**; 
also in his collection of letters* is a letter from Theodoric to Boethius 
containing the words, "for in your translations ... Ntcomachus the 
arithmetician, and Euclid the geometer^ are heard in the Ausonian 
tongue" The so-called Geometry of Boethius which has come down 
to us by no means constitutes a translation of Euclid The MSS. 
variously give five, four, three or two Books, but they represent only 
two distinct compilations, one normally in five Books and the other 
in two. Even the latter, which was edited by Friedlein, is not 
genuine*, but appears to have been put together in the nth c», from 
various sources. It begins with the definitions of EucL I,, and in these 
are traces of perfectly correct readings which are not found even in 
the MSS. of the loth c, but which can be traced in Proclus and other 
ancient sources ; then come the Postulates (five only), the Axioms . 
(three only), and after these some definitions of Eucl 11., 11 L, iv. 
Next come the enunciations of Eucl I., of ten propositions of Book 11,, 
and of some from Books llL, IV., but always without proofs ; there 
follows an extraordinary passage which indicates that the author will 
now give something of his own in elucidation of Euclid, though what ' 
follows is a literal translation of the proofs of EucL I. I — 3, This 
latter passage, although it affords a strong argument against the 
genuineness of this part of the work, shows that the Pseudoboethius 
had a Latin translation of Euclid from which he extracted the three 
propositions. 

Curtze has reproduced, in the preface to his edition of the trans- 
lation by Gherard of Cremona of an-Nairizf s Arabic commentary on 
Euclid, some interesting fragments of a translation of Euclid taken 
from a Munich MS. of the loth c. They are on two leaves used 
for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 
2*^ 757) atr>d consist of portions of Eucl, L 37, 38 and IL 8, translated 
literally word for word from the Greek text The translator seems to 
have been an Italian (c£ the words '' capitolo nono " used for the ninth 
prop, of Book II.) who knew very little Greek and had moreover little 
mathematical knowledge. For example, he translates the capital letters 
denoting points in figures as if they were numerals : thus tA ABF, 

* The frugment wms deciphered by W. Stndemmid, who oommimicated his resalts to 
Cantor. 

* CaiBodorius, Varioi^ i. 45, p. 40, 13 ed. Mommieii. 

* See etpedaUy Weissenbom in AbkatuUumgm mr Getck, d. Math. II. p. 185 iq.; 
1 Heibcsg in Pkilol^gus^ XLUI. p. 507 sq. ; Cantor, ig, p. 580 sq. 



CH. viii] TRANSLATIONS AND EDITIONS 93 

AEZ is translated ''que primo secundo et tertio quarto quinto et 
scptimo," T becomes "tricentissimo " and so on. The Greek MS. which 
he used was evidently written in uncials, for AEZ6 becomes in one 
place " quod autem septimo nono," showing that he mistook AE for 
the particle &', and koI 6 2TU is rendered ''sicut tricentissimo et 
quadringentissimo/' showing that the letters must have been written 
KAIOCTU. 

The date of the Englishman Athelhard (iEthelhard) is approxi- 
mately fixed by some remarks in his work Perdiffidles Quaestiones 
Naturales which, on the ground of the personal allusions they contain, 
must be assigned to the first thirty years of the 12th c* He wrote a 
number of philosophical works. Little is known about his life. He 
is said to have studied at Tours and Laon, and to have lectured at the 
latter school. He travelled to Spain, Greece, Asia Minor and Egypt, 
and acquired a knowledge of Arabic, which enabled him to translate 
from the Arabic into Latin, among other works, the Elements of 
Euclid. The date of this translation must be put at about 11 20. 
MSB. purporting to contain Athelhard's version are extant in the 
British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 
47 and Ball. Coll. 257 of 12th c), Nurnberg (Johannes Regiomontanus' 
copy) and Erfurt 

Among the very numerous works of Gherard of Cremona (i 1 14 — 
1 1 87) are mentioned translations of ** 15 Books of Euclid" and of the 
Data}. Till recently this translation of the Elements was supposed to 
be lost; but Axel Anthon Bjombo has succeeded (1904) in discovering 
a translation from the Arabic which is different from the two others 
known to us (those by Athelhard and Campanus respectively), and 
which he, on grounds apparently convincing, holds to be Gherard's. 
Already in 1901 Bjombo had found Books x. — xv. of this translation 
in a MS. at Rome (Codex R^inensis lat. 1268 of 14th c.)'; but three 
years later he had traced three MSB. containing the whole of the same 
translation at Paris (Cod. Paris. 7216, 15th c), Boulogne-sur-Mer 
(Cod. Bononiens. 196, 14th c), and Bruges (Cod. Brugens. 521, 14th c), 
and another at Oxford (Cod. Digby 174, end of 12th c.) containing a 
fragment, XI. 2 to XIV.* The occurrence of Greek words in tfiis 
translation such as rombus^ romboides (where Athelhard keeps the 
Arabic terms), ambligonitis, orthogonius, gnomo^ fyramis etc., show 
that the translation is independent of Athelhard's. Gherard appears 
to have had before him an old translation of Euclid from the Greek 
which Athelhard also often followed, especially in his terminology, 
using it however in a very different manner. Again, there are some 
Arabic terms, e.g. meguar for axis of rotation^ which Athelhard did not 
use, but which is found in almost all the translations that are with 
certainty attributed to Gherard of Cremona; there occurs also the 

» Cantor, Gesch, d. Math, i„ p. 006. 

* Boncompagni, Delia vita e delU cfere di Gherardo Crewunuse^ Rome, 1851, p. 5. 

* Described in an appendix to Studien iiber Menelaos^ Sphdrik {Abhanduungm %ur 
Gtschuhte der mathematischen IVissenschafien, xiv., 1901). 

* Sec Bibliotheca Mathematica, vig, 1905-^, pp. 141—8. 



94 INTRODUCTION [ck. viii 

expression "superficies equidjstantium latcnim et rectomm angulorum/' 
found also in Gherard's translation of an-NairM, where Athelhard says 
"parallelogrammum rectan^lum/' The translation is much clearer 
than Athelhard's: it is neither abbreviated nor "edited'' as Athelhard's 
appears to have been ; it is a word-for*word translation of an Arabic 
MS. containing a revised and critical edition of Thabit^s version. It 
contains several notes quoted from Thabit himself {Thebit dixii\ e.g. 
about alternative proofs etc* which Thabit found ''in another Greek 
MS./* and is therefore a further testimony to Thabit s critical treatment 
of the text after Greek MSS. The new editor also added critical 
remarks of his own* e.g. on other proofs which he found in other 
Arabic versions, but not in the Greek; whence it is clear that he 
compared the Thabit version before him with other versions as care- 
fully as Thabit collated the Greek MSS. Lastly, the new editor speaks 
of ''Thebit qui transtulit hunc librum in arabicam linguam" and of 
"translatioThebit," which may tend to confirm the statement of al-Qifti 
who credited Thabit with an independent translation, and not (as the 
Fihrist does) with a mere improvement of the version of Ishaq b, 
^unain. 

Gherard's translation of the Arabic commentary of an-NatiizI on 
the first ten Books of the EUments was discovered by Maximitian 
Curtze in a MS. at Cracow and published as a supplementary volume 
to Heiberg and Menge's Euclid^: it will often be referred to in this 
work. 

Next in chronological order comes Johannes Campanus (Cam pane) 
of Novara. He is mentioned by Roger Bacon (12(4-1294) as a 
prominent mathematician of his time*, and this indication of his date 
is confirmed by the fact that he was chaplain to Pope Urban IV, who 
was Pope from 1261 to 1281*. His most important achievement was 
his edition of the Elements including the two Books XIV. and XV. 
which are not Euclid's. The sources of Athelhard's and Campanus' 
translations, and the relation between them, have been the subject of 
much discussion, which does not seem to have led as yet to any 
definite conclusion. Cantor (III, p. 91) gives references^ and some 
particulars. It appears that there is a MS. at Munich (Cod. lat. Mon. 
1 3021) written by Sigboto in the 12th c at Priifning near Regensburg, 
and denpted by Curtze by the letter R, which contains the enunciations 
of part of Euclid. The Munich MSS. of Athelhard and Campanus' 
translations have many enunciations textually identical with those in 
R, so that the source of all three must, for these enunciations, have 

' AnariHi in decern HbrM friores EUmenUrum EucUdis Commentarii ex inter^niaUcmi 
Gherardi Cremonensis in codice Cracaviensi 569 servata edidit Maximiliaous Cunze, Leiptig 
(Teubner), 1899. 

* Cantor, Hi, p. 88. 

' Tiraboschi, Storia della leiteratura ita/iana, IV. 145—160. 

* H. V^eissenbom in ZeUschrift fiir Math, «. Pkysik^ xxv., Supplement, pp. T43— 166, 
and in his monograph, Die Obenettungen des Ettkliddurck Campano und ZambertiXi^i) ; 
Max. Curtze in FhtloUgiiche Rundschau (1881), I. pp. 943^950, and in Jdhresbericht iiAtr 
die Forischritte der classischen AUerihumswissemchaft^ XL. (1C64, in.) pp. 19 — 39 ; Heibeig 
in ZaischriftJUr Math, u. Physik^ XXXV., hist.-litt Abth., pp. 48—58 and pp. 81—6. 






CH. viii] TRANSLATIONS AND EDITIONS 95 

been the same; in others Athelhard and Campanus diverge com- 
pletely from R, which in these places follows the Greek text and is 
therefore genuine and authoritative. In the 32nd definition occurs the 
word " elinuam," the Arabic term for " rhombus," and throughout the 
translation are a number of Arabic figures. But R was not translated 
from the Arabic, as is shown by (among other things) its close 
resemblance to the translation from Euclid given on pp. 377 sqq. of 
the Gromatici Veteres and to the so-called geometry of Boethius. The 
explanation of the Arabic figures and the word " elinuam " in Def. 32 
appears to be that R was a late copy of an earlier original with 
corruptions introduced in many places ; thus in Def. 32 a part of the 
text was completely lost and was supplied by some intelligent copyist 
who inserted the word "elinuam," which was known to him, and also 
the Arabic figures. Thus Athelhard certainly was not the first to 
translate Euclid into Latin ; there must have been in existence before 
the nth c. a Latin translation which was the common source of R, 
the passage in the Gromatici^ and '' Boethius." As in the two latter 
there occur the proofs as well as the enunciations of L i — 3, it is 
possible that this translation originally contained the proofs also. 
Athelhard must have had before him this translation of the 
enunciations, as well as the Arabic source from which he obtained his 
j proofs. That some sort of translation, or at least fragments of one, 
were available before Athelhard's time even in England is indicated 
by some old English verses^ : 

*'The clerk Euclide on this wyse hit fonde 
Thys craft of gemetry yn Egypte londe 
Yn Egypte he tawghte hyt ful wyde, 
In dyvers londe on every syde. 
Mony erys afterwarde y understonde 
Yer that the craft com ynto thys londe. 
Thys craft com into England, as y yow say, 
Yn tyme of good kyng Adelstone's day," 

which would put the introduction of Euclid into England as far back 
as 924-940 A.D. 

We now come to the relation between Athelhard and Campanus. 
That their translations were not independent, as Weissenborn would 
have us believe, is clear from the fact that in all MSS. and editions, 
apart from orthographical differences and such small differences as 
are bound to arise when MSS. are copied by persons with some 
knowledge of the subject-matter, the definitions, postulates, axioms, 
and the 364 enunciations are word for word identical in Athelhard 
and Campanus ; and this is the case not only where both have the 
same text as R but where they diverge from it. Hence it would seem 
that Campanus used Athelhard's translation and only developed the 
proofs by means of another redaction of the Arabian Euclid. It is 
true that the difference between the proofs of the propositions in the 
two translations is considerable; Athelhard's are short and com- 

^ Quoted by Halliwell in Rara Maihematica (p. 56 note) from MS. Bib. Reg. Mas. Brit. 
17 A. I. f. 2^—3. 



96 INTltOmTCTICni 



pressed, Campanus' cleamr and mote c o mp lete, iolkMrlaff tile Gnei 
text more closely, thoadi still at aome diftaiioe» Ftetibtt^: lb 
arrangement in the two is diflerent; in AtbeUianl the praoft itgitM i ty | j 
precede the enunciation^ Dunpanos §oXkmm tlie mual ctdm: ft iaa 
question how far the differencet in the pnxA, and certain additiwiJtt 
each, are due to the two transtatoia tfaantdves or go back to AmUc 
originals. The latter supposition seems to Cocttt and Cantor the 
more probable one. Curtae's general view of tiie relation of Qininan n s 
to Athelhard is to the effect that Adidhard's tinnslalion was nwiMllj 
altered, from the form in iHiidi it appears in the two Emmt Wm. 
described by Weissenbom, fay socoesahre copyists and ciwiwi i mta tor s 
wAo had Arabic originmU m^itm tlmm^ w^ it^ took ti» faraa irilidi 
Campanus gave it and in which it was pnUished In soppoit of ttis 
view Curtze refers to Rq;iomontaniis*copy of the AtfadharcMranipanns 
translation. In R^omontanus' own prc&ce the title is given, 'Md 
this attributes the translation to Atfidhard; bit^ wiiite diia copy 
agrees almost exactly wiUi Athelhard In Book L, yet, in places wtoa 
Campanus is more lengthy, it has similar additions, and in the Isler 
Books, especially from Book Ul. onwards, agrees ahsdittd^ wfth 
Campanus; R^iomontanu% too^ himself implies tinrt, fhrnMh A 
translation was Athelhard's, Campanus had reidsed it; fcr ha !■ 
notes in the margin such as the following, **Campani est hec^" ''dnfako 
an demonstret hie Campanus " etc. * 

We come now to the printed editions of tlie udiole or of portfams 
of the ElemenU, This is not the idace for a complete bihlkiipaplqr» : 
such as Riccardi has attempted in his valuable memoir issued in five 
parts between 1887 and 1893, which makes a large book in itself ^ 
I shall confine myself to saying something of the most noteworthy 
translations and editions. It will be convenient to give first the Latin 
translations which preceded the publication of the editio prinups of 
the Greek text in 1 533, next the most important editions of the Greek 
text itself, and after them the most important translations arranged 
according to date of first appearance and languages, first the Latin 
translations after 1533, then the Italian, German, French and English 
translations in order. 

It may be added here that the first allusion, in the West, to the 
Greek text as still extant is found in Boccaccio's commentary on the 
Divina Commedia of Dante*. Next Johannes Regiomontanus, who 
intended to publish the Elements after the version of Campanus, but 
with the latter^s mistakes corrected, saw in Italy (doubtless when 
staying with his friend Bessarion) some Greek MSS. and noticed how 
far they differed from the Latin version (see a letter of his written in | 
the year 147 1 to Christian Roder of Hamburg)*. 

^ Saggw di una Bibliogrt^ EucUdea^ memoria del Prof. Pietio Riccardi (Bologiui, ! 
1887, 1888, 1890, 1893). ^ 

" I. p. 404. 

* Pabliahed in C. T. de Mail's fitmorabilia BibUatiucarum NorimUrgpumm^ Pttrt I. ' 
p. 190 iqq. 



i 

i CH. viii] TRANSLATIONS AND EDITIONS 97 



I 



I. Latin translations prior to 1533. 



I 1482. In this year appeared the first printed edition of Euclid, 

which was also the first printed mathematical book of any import- 
ance. This was printed at Venice by Erhard Ratdolt and contained 
Campanus' translation \ Ratdolt belonged to a family of artists at 
Augsburg, where he was born about 1443. Having learnt the trade 
of printing at home, he went in 1475 to Venice, and founded there a 
famous printing house which he managed for 1 1 years, after which he 
returned to Augsburg and continued to print important books until 
1 5 16. He is said to have died in 1528. Kastner' gives a short 
description of this first edition of Euclid and quotes the dedication to 
Prince Mocenigo of Venice which occupies the page opposite to the 
first page of text. The book has a margin of 2^ inches, and in this 
margin are placed the figures of the propositions. Ratdolt says in 
his dedication that at that time, although books by ancient and 
modern authors were printed every day in Venice, little or nothing 
mathematical had appeared : a fact which he puts down to the diffi- 

Iculty involved by the figures, which no one had up to that time 
succeeded in printing. He adds that after much labour he had 
discovered a method by which figures could be produced as easily as 
letters*. Experts are in doubt as to the nature of Ratdolt's discovery. 
Was it a method of making figures up out of separate parts of figures, 
straight or curved lines, put together as letters are put together to 
make words > In a life of Joh. Gottlob Immanuel Breitkopf, a con- 
temporary of Kastner's own, this member of the great house of 
Breitkopt is credited with this particular discovery. Experts in that 
^same house expressed the opinion that Ratdolt's figures were wood- 
'cuts, while the letters denoting points in the figures were like the 
other letters in the text ; yet it was with carved wooden blocks that 
printing b^an. If Ratdolt was the first to print geometrical figures, 
it was not long before an emulator arose ; for in the very same year 

IMattheus Cordonis of Windischgratz employed woodcut mathematical 
figures in printing Oresme*s De laHtudinibus\ How eagerly the 
opportunity of spreading geometrical knowledge was seized upon is 
proved by the number of editions which followed in the next few 
years. Even the year 1482 saw two forms of the book, though they 
only differ in the first sheet Another edition came out in i486 
{Ulmae, apud lo, Regerutn) and another in 149 1 {Vincentiae per 

^ Curtze (An-NairizI, p. xiii) reproduces the heading of the first page of the text as 

follows (there is no title-pa^e) : PreclarifTimu opus elemento^ Eucliois megarefis vna cu 

cOmentis Camponi pfpicacifnini in arte geometric incipit felicit*, after which the definitions 

begin at once. Other copies have the shorter heading : Preclarissimus liber elementonim 

Euclidis perspicacissimi : in artem Geometric incipit quam foelicissime. At the end stands 

I the following : C Opus elementoru euclidis megarenf is in geometria arte Jn id quoq} Campani 

I pfpicaciffinu C6mentationes finiut. Erhardus ratdolt Augustensis imprefTor folertiffimus . 

; venetijs iropreirit . Anno ialutis . M.cccc.lxxxij . Octauis . Calefi . Jufi . Lector . Vale. 

t' Kastner, GtschUhte dtr Maihematik^ I. p. 189 sqq. See also Weissenbom, Die Obersett- 
ungen des EukHd durch Campano und Zamoertit pp. i — 7. 
' "Mea industria non sine maximo labore efieci vt qua fturilitate litterarum elementa 
imprimuntur ea etiam geometrice figure conficerentur." 

* Curtze in Zeiischriftfiir Math, u, Physik, XX., hist.-litt. AbUi. p. 58. 

H. E. 7 



I 



r 

98 INTRODUCTION [cH.Tin 

Leonardum de BasUea ei GuHditmm d$ P^^i^\ but witiiout Ae dedi- 
cation to Mocenigo who had died in the meantime (1485)^ If Cam- 
panus added anything of his own, his additions are at all events not 
distinguished by any difference of t^pe or oAerwise; the enunciations 
are in large type, and the rest is prmted continuously in smaller type. 
There are no superscriptions to particular passages such as Ettduks 
ex Campano^ Campanus^ Cm^ami addiiio^ or Omi$mU amMtoHo^ which 
are found for the first time in the Paris edition of 1516 giving 
both Campanus' version and that of Zamberti Qwesently to be men- 
tioned). 

1 501. G. Valla included in his enc3^opaedic work De expiUndis 
et fugiendis rebus published in this year at Venice (iVk eudibus Atdi 
Romant) a number of propositions with proofs and scholia translated 
from a Greek MS. which was once in hiis possession (cod Mutin. ni 
B, 4 of the isthc). 

1 505. In this year Bartolomeo Zamberti (Zambertus) brought out 
at Venice the first translation, from the Greek text, of the whole of tlie 
Elements. From the titled as well sis from his prefaces to the CaUptriem 
and DaUif with their allusions to previous translators ** who take some 
things out of authors, omit some, and change some," or ** to that most 
barl^rous translator " who filled a volume purporting to be Euclid's 
''with extraordinary scarecrows, nightmares and phantasies," one object 
of Zamberti's translation b dear. His animus against Campanus 
appears also in a number of notes, e.g. when he condemns the terms 
''helmuain" and '^ helmuariphe "* used by Campanus as barbarous^ 
un-Latin etc, and when he is roused to wrath by Campanus' unfortu- I 
nate mistranslation of v. Def. 5. He does not seem to have had the > 
penetration to see that Campanus was translating from an Arabic, 
and not from a Greek, text Zamberti tells us that he spent 
seven years over his translation of the thirteen Books of the 
Elements. As he seems to have been bom in 1473, and the Elements 
were printed as early as 1500, though the complete work (including the 
Phaenomena^ Optica^ Catoptrica^ pata etc.) has the date 1505 at the 
end, he must have translated Euclid before the age of 3a Heiberg 
has not been able to identify the MS. of the Elements which Zamberti 
used ; but it is clear that it belonged to the worse class of MSS., since 
it contains most of the interpolations of the Theonine variety. Zam- 
berti, as his title shows, attributed th^ proofs to Theon. 

1509. As a counterblast to Zamberti, Luca Paciuolo brought out 
an edition of Euclid, apparently at the expense of Ratdolt, at Venice f 
{per Pagamnum de Paganinii), in which he set himself to vindicate 
dampanus. The title-page of this now very rare edition' b^ns thus : | 
*'The works of Euclid of Megara, a most acute philosopher and without ' 

^ The title begins thus: *'Eiiclidis megaresis philosophi platonicj mathematicaniml 
discipliiuurum Janitoris : Habent in hoc volumine quicunque ad matbematicaro substantiam . 
aspirant : elementomm libros xiij cum expositione Theonis insignis mathematici. quibus, 
multa quae deerant ex lectione graeca sumpta addita sunt nee non plarima peniena et 
praepostere: voluta in Campani interpretatione : ordinata digesta et castigata sant etc." 
For a description of the book see Weissenbom, p. is sqq. 
. ' See Weissenbom, p. 30 sqq. 



CH. viii] TRANSLATIONS AND EDITIONS 99 

question the chief of all mathematicians, translated by Campanus their 
most faithful interpreter'' It proceeds to say that the translation had 
been, through the fault of copyists, so spoiled and deformed that it 
could scarcely be recognised as Euclid. Luca Paciuolo accordingly 
has polished and emended it with the most critical judgment, has 
corrected 129 figures wrongly drawn and added others, besides supply- 
ing short explanations of difficult passages. It is added that Scipio 
V^ius of Milan, distinguished for his knowledge '' of both languages'* 
(le. of course Latin and Greek), as well as in medicine and the more 
sublime studies, had helped to make the edition more perfect. Though 
Zamberti is not once mentioned, this latter remark must have refer- 
ence to Zamberti's statement that his translation was from the Greek 
text ; and no doubt Zamberti is aimed at in the wish of Paciuolo's 
" that others too would seek to acquire knowledge instead of merely 
showing off, or that they would not try to make a market of the 
things of which they are ignorant, as it were (selling) smoke*." 
Weissenborn observes that, while there are many trivialities in Paci- 
uolo's notes, they contain some useful and practical hints and explana- 
tions of terms, besides some new proofs which of course are not 
difficult if one takes the liberty, as Paciuolo does, of diverging from 
Euclid's order and assuming for the proof of a proposition results not 
arrived at till later. Two not inapt terms are used in this edition to 
describe the figures of ill. 7, 8, tihe former of which is called the 
gooses foot {pes anseris\ the second the peacock's tail {catula pavonis), 
Paciuolo as the castigator of Campanus' translation, as he calls himself, 
failed to correct the mistranslation of v. Def 5'. Before the fifth 
Book he inserted a discourse which he gave at Venice on the 
15th August, 1508, in S. Bartholomew's Church, before a select 
audience of 500, as an introduction to his elucidation of that Book. 
1 5 16. The first of the editions giving Campanus' and Zamberti's 
translations in conjunction was brought out at Paris (i« offidna Henrici 
Stephani e regione scholae Decretorum). The idea that only the enun- 
ciations were Euclid's, and that Campanus was the author of the proofs 
in his translation, while Theon was the author of the proofs in the Greek 
text, reappears in the title of this edition; and the enunciations of the 
added Books xiv., xv. are also attributed to Euclid, Hypsicles being 
credited with the proofs'. The date is not on the title-page nor at the 

^ "Atque utinam et alii cognoscere vellent nod ostentare ant ea quae nesdunt velati 
fumum venditare non conarentar.'* 

' Campanus' translation in Ratdolt's edition is as follows: "Quantitates quae dicnntur 
oontinuam habere proportionaliutem, sunt, quarum equ^ multiplicia aut equa sunt aut 
eou^ sibi sine interruptione addunt aut minuunt " ( !), to which Campanus adds the note : 
^ Continue proportionalia sunt quorum omnia multiplicia equalia sunt continue proportionalia. 
Sed noluit ipsam diffinitionem proponere sub hac forma, quia tunc difiiniret idem per idem, 
aperte (? a parte) tamen rei est istud cum sua diffinitione convertibile." 

' **£uclidis Megarensis Geometricorum Elementorum Libri xv. Campani Galli trans- 
alpini in eosdem commentariorum libri xv. Theonis Alexandrini Bartholomaeo Zambertd 
Veneto interprete, in tredecim priores, commentationum libri xiii. Hypsiclis Alexandrini in 
duos posteriores, eodem Bartholomaeo Zamberto Veneto interprete, commentariorum libri n." 
On the last page (161) is a similar sutement of content, but with the difference that the 
expression **ex Campani... deinde Theonis... et Hypsiclis... /nu/iM^t^Kr." For description 
see Weissenborn, p. 56 sqq. 

7-^2 



loo INTRODUCTION \ ' [€& vm 

end, but the letter of dedication to Franjois Briconnet by Jacques 
Leftvre is dated the day after the Epiphany, 151& The figures are 
in the margin. The arrangement of the propositions is as follows : 
first the enunciation with the beading Eudides ex Campano^ then the 
proof with the note Campanus^ and after that, as Campani additio^ any 
passage found in the ^ition of Campanus' translation but not in the 
Gr^k text ; then follows the text of the enunciation translated from 
the Greek with the heading Eudides ^x Zambtrto, and lastly the proof 
headed Theo ex ZambertQ, There are separate figures for the two proofs. 
This edition was reissued with few changes in 1537 and 1546 at Basel 
(fipud lokaumm H€rvagium\ but ivith the addition of the Pkojenontitta^ 
Optica^ Catoptrka etc For the edition of 1537 the Paris edition of 
1516 was collated with *'a Greek copy" (as the preface says) by 
Christian Herlinj professor of mathematical studies at Strassburg, 
who however seems to have done no more than correct one or two 
passages by the help of the Basel editw princeps (1533), and add the 
Greek word in cases where Zamberti's translation of it seemed unsuit- 
able or inaccurate. ' 

We now come to # , t, 

. . T 
IL Editions of the Greek text, » •* 

1 533 is the date of the editw prinaps, the title-page of which reads 
as follows: 

ETKAEIAOT STOIXEinN BIBA>- lEK 

EK TON eEONOZ 1TN0Y2I0R 
£«'? ToO avTov TO TrpmroVf e^^^dnop TlpoicXov 0ifi\* o* 

Adiecta praefatiuncula in qua de disctplinis # 

Mathematicis nonnihiL 

BASILEAE APVD lOAN, HERVAGIVM ANNO 

M^aXXXIlL MENSE SEPTEMBRL 

The editor was Simon Grynaeus the elder (d* 1541), who, after 
working at Vienna and Ofen, Heidelberg and Tubingen, taught last 
of all at Basel, where theology was his main subject. His "prae- 
fatiuncula" is addressed to an Englishman, Cuthbert Tonstall (1474- 
1559)9 ^ho> having studied first at Oxford, then at Cambridge, where 
he became Doctor of Laws, and afterwards at Padua, where in addi- 
tion he learnt mathematics — mostly from the works of Regiomontanus 
and Paciuolo — wrote a book on arithmetic* as **a farewell to the 
sciences," and then, entering politics, became Bishop of London and 
member of the Privy Council, and afterwards (i 530) Bishop of Durham. 
Grynaeus tells us that he used two MSS. of the text of the Elements, 
entrusted to friends of his, one at Venice by " Lazarus Bayfius " 
(Lazare de BaTf, then the ambassador of the King of France at Venice), 
the other at Paris by " loann. Rvellius " (Jean Ruel, a French doctor 
and a Greek scholar), while the commentaries of Proclus were put at 

^ De arti tufputamdiiiM qmatuor. 



1 



CH. viii] TRANSLATIONS AND EDITIONS ..;v loi 

the disposal of Grynaeus himself by "loann. Claymundus** j;t*Qxford. 
Heiberg has been able to identify the two MSS. used for* the** text ; 
they are (i) cod. Venetus Marcianus 301 and (2) cod. Paris. g^.*2343 
of the 1 6th c, containing Books I. — XV., with some scholia wHicJ^* are 
embodied in the text. When Grynaeus notes in the margfff \t^e 
readings from "the other copy," this "other copy" is as a rule** the 
Paris MS., though sometimes the reading of the Paris MS. is taken, 
into the text and the " other copy " of the margin is the Venice Mfi* 
Besides these two MSS. Grynaeus consulted Zamberti, as is shown by^* 
a number of marginal notes referring to " Zampertus " or to " latinum ' 
exemplar" in certain propositions of Books IX. — XL When it is con- 
sidered that the two MSS. used by Grynaeus are among the worst, it 
is obvious how entirely unauthoritative is the text of the editio prinups. 
Yet it remained the source and foundation of later editions of the 
Greek text for a long period, the editions which followed being 
designed, not for the purpose of giving, from other MSS., a text more 
nearly representing what Euclid himself wrote, but of supplying a 
handy compendium to students at a moderate price. 

1536. Orontius Finaeus (Oronce Fine) published at Paris {apud 
Simonetn Colinaeuni) "demonstrations on the first six books of Euclid's 
elements of geometry," " in which the Greek text of Euclid himself is 
inserted in its proper places, with the Latin translation of Barth. 
Zamberti of Venice," which seems to imply that only the enunciations 
were given in Greek. The preface, from which Kastner quotes S says 
that the University of Paris at tfiat time required, from all who 
aspired to the laurels of philosophy, a most solemn oath that they 
had attended lectures on the said first six Books. Other editions of 
Fine's work followed in 1544 and 1551. 

1545. The enunciations of the fifteen Books were published in 
Greek, with an Italian translation by Angelo Caiani, at Rome {apud 
Antonium Bladum Asulanum), The translator claims to have cor- 
rected the books and " purged them of six hundred things which did 
not seem to savour of the almost divine genius and the perspicuity of 
Euclid'." 

1 549. Joachim Camerarius published the enunciations of the first 
six Books in Greek and Latin (Leipzig). The book had a preface by 
Rhaeticus, a pupil of Copernicus, bom at Feldkirch in the Vorarlberg 
1514, died 1576. Another edition with proofs of the propositions 
of the first three Books was published by Moritz Steinmetz in 1577 
(Leipzig). 

1550. loan. Scheubel published at Basel (also per loan. Her- 
vagiunt) the first six Books in Greek and Latin " together with true 
and appropriate proofs of the propositions, without the use of letters " 
(to denote points in the figures, the various straight lines and angles 
being described in words*). 

1557 (also 1558). Stephanus Gracilis published another edition 
(repeated 1 573, 1 578, 1 598) of the enunciations (alone) of Books I. — xv. 

* Kiistner, I. p. a6o. ' Heiberg, vol. v. p. cvii. * K&stner, i. p. 359. 



% 



...... INTRODUCTION i W^[ch. viii 

in Grcetf^tid Latin at Paris (afiud GuUelmum Caviilat\ He remarks 
tn thtf'preTace that for want of time he had changed scarcely anything 
/in 5doJcs L — VI., but m the remaining Books he had emended what 
[seemed obscure or inelegant in the Latin translation, while he had 
' ad6pCed in its entirety the translation of Book X. by Pierre Mondor<i 
{^P^lrus Montaureus), published separately at Paris in 1551- Gracilis 
also added a few ** scholia." 

**; 1564, In this year Conrad Dasypodius (Rauchfuss), the inventor 
and maker of the clock in Strassburg cathedral, similar to the present 
one, which did duty from 1571 to 1789, edited (Strassburg, Chn 
Mylius) (i) Book L of the Elements in Greek and Latin with scholia, 
(2) Book IL in Greek and Latin with Barlaain*s arithmetical version 
of Book IL, and (3) the muncmticns of the remaining Books HI. — XllL 
Book L was reissued with "vocabula quaedam geometrica" of Heron, 
the enunciations of all the Books of the Eiemenis, and the other works 
of Euclid, all in Greek and Latin. In the preface to (l) he says that it 
had been for twenty-six years the rule of his school that all who were 
promoted from the classes to public lectures should learn the first 
Book, and that he brought it out, because there were then no longer 
any copies to be had, and in order to prevent a good and fruitful 
regulation of his school from falling through. In the preface to the 
edition of 1571 he says that the first Book was generally taught in all 
gymnasia and that it was prescribed in his school for the first class. 
In the preface to (3) he tells us that he published the enunciations of 
Books IIL — XIII, in order not to leave his work unfinished, but that, as 
it would be irksome to carry about the whole work of Euclid in 
extenso, he thought it would be more convenient to students of 
geometry to learn the Eiefnmts if they were compressed into a smaller 
book. 

t62a Henry Briggs (of Briggs* logarithms) published the first 
six Books in Greek with a Latin translation after Command in us, 
"corrected in many places" (London, G. Jones). 

1703 is the date of the Oxford edition by David Gregory which, 
until the issue of Heiberg and Menge's edition, was still the only 
edition of the complete works of Euclid ^ In the Latin translation 
attached to the Greek text Gregory says that he followed Comman- 
dinus in the main, but correct^ numberless passages in it by means 
of the books in the Bodleian Library which belonged to Edward 
Bernard (1638- 1696), formerly Savilian Professor of Astronomy, who 
had conceived the plan of publishing the complete works of the ancient 
mathematicians in fourteen volumes, of which the first was to contain 
Euclid's Elements L — KV» As regards the Greek text, Gregory tells us 
that he consulted, as far as was necessary, not a few MSS. of the better 
sort, bequeathed by the great Savile to the University, as well as the 
corrections made by Savile in his own hand in the margin of the Basel 
edition. He had the help of John Hudson, Bodley s Librarian, who 

^ ETEABtAOT tA ZCIZOMBKA. EucHdis q^e sapersunt omnut. Ex re<^nnotie 
D«TidU Grcgorii M.D, Astronomiae Professoris SavilUnt ci R.S,S, Oxoniac, c Tbeairo 
Sheldoniuto, An. Dom. iiDcciti. 



I 



CH. viii] TRANSLATIONS AND EDITIONS 103 

punctuated the Basel text before it went to the printer, compared the 
Latin version with the Greek throughout, especially in the Elements 
and Data, and, where they differed or where he suspected the Greek text, 
consulted the Greek MSS. and put their readings in the margin if 
they agreed with the Latin and, if they did not agree, affixed an 
asterisk in order that Gregory might judge which reading was geo- 
metrically preferable. Hence it is clear that no Greek MS., but the 
Basel edition, was the foundation of Gregory's text, and that Greek 
MSS. were only referred to in the special passages to which Hudson 
called attention. 

1 8 14-18 1 8. A most important step towards a good Greek text 
was taken by F. Peyrard, who published at Paris, between these years, 
in three volumes, the Elements and Data in Greek, Latin and French ^ 
At the time (1808) when Napoleon was having valuable MSS. selected 
from Italian libraries and sent to Paris, Peyrard managed to get two 
ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 
204 was also at Paris at the time, but all three were restored to their 
owners in 18 14). Peyrard noticed the excellence of Cod. Vat. 190, 
adopted many of its readings, and gave in an appendix a conspectus 
of these readings and those of Gregory's edition ; he also noted here 
and there readings from Vat. .1038 and various Paris MSS. He there- 
fore pointed the way towards a better text, but committed the error 
of correcting the Basel text instead of rejecting it altogether and 
starting afresh. 

1 824- 1 825. A most valuable edition of Books I. — ^vi. is that of 
J. G. Camerer (and C. F. Hauber) in two volumes published at 
Berlin". The Greek text is based on Peyrard, although the Basel 
and Oxford editions were also used. There is a Latin translation 
and a collection of notes far more complete than any other I have 
seen and well nigh inexhaustible. There is no editor or commentator 
of any mark who is not quoted from ; to show the variety of important 
authorities drawn upon by Camerer, I need only mention the following 
names : Proclus, Pappus, Tartaglia, Commandinus, Clavius, Peletier, 
Barrow, Borelli, Wallis, Tacquet, Austin, Simson, Playfair. No words 
of praise would be too warm for this veritable encyclopaedia of 
information. 

1825. J. G. C. Neide edited, from Peyrard, the text of Books 
I. — VI., XI. and XII. {Halts Saxoniae). 

1826-9. The last edition of the Greek text before Heiberg's is 
that of E. F. August, who followed the Vatican MS. more closely 
than Peyrard did, and consulted at all events the Viennese MS. 
Gr. 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains 
Books I. — XIII. 

^ Etulidis quae iupersunt, Les (Euvres cTEuclide^ en Grec, en Laiin et en Fran^ais 
d^aprls un manuserit tris-ancien, qui itait restiinconnu jusqu^h nos jours. Par F. Peyrard. 
Ouvrage approuv^ par TlDStitut de France (Paris, chez M. Patris). 

* Euclidis elementorum Ubri sex pricres graece et laiine commentarU e scriptis veterum ac 
ncentiorum mathematuorum et PJUuieren maxime i/Zf/j/m/f (Berolini, sumptibus G. Reimeri). 
Tom. I. 1834; torn. n. 1835. 






I 



«4 INTRODUCTION r [ciu vui 

III, Latin versions or commentaries after 153?*""" 

1545, Petrus Ramus (Pierre de la Ratnte* 15 15-1 572) is credited 
with a translation of Euclid which appeared in 1545 and again in 
1 549 at Parish Ramus, who was more rhetorician and logician thanj 
geometer, also published in his ScAolae ma fA^malicae {i ^$g,Fr^nk{uTtl\ 
1569, Basel) what amounts to a series of lectures on Euclid's Ekmtnts^^ 
in which he criticises Euclid's arrangement of his propositions, the; 
definittons, postulates and axioms^ all from the point of view of logiaj 

1557. Demonstrations to the geometrical Elements of Euclid, six' 
Books, by Peletarius (Jacques Peleticr), The second edition (1610) 
contained the same with the addition of the "Greek text of Euclid**; 
but only the enunciatmis of the propositions, as well as the defini- 
tions etc*j are given in Greek (with a Latin translation), the rest is 
in Latin only* He has some acute observations, for instance about 
the "angle" of contact \ 

1S59» Johannes Buteo, or Borrel (1492-1572), published in anj 
appendix to his book De quadraiura circuli some notes *'on the errors^ 
of Campanus, Zambertus, Orontius, Peletarius^ Pena, interpreters of 
Euclid/* Buteo in these notes proved, by reasoned argument based 
on original authorities, that Euclid himself and not Theon was the 
author of the proofs of the propositions* 

1566. Franciscus Flussates Candalla (Francois de Foix, Comte de 
Candale, 1502-1594) "restored" the fifteen Books, following, as he 
says, the terminology of Zamberti s translation from the Greek, but 
drawing, for his proofs, on both Campanus and Theon (i,e, Zamberti) 
except where mistakes in tliem made emendation necessary. Other 
editions followed in 1578, 1602, 1695 (in Dutch). 

1572. The most important Latin translation is that of Com- 
mandinus (i 509-1 575) of Urbino, since it was the foundation of most 
translations which followed it up to the time of Peyrard» including 
that of Simson and therefore of those editions, numerous in England, 
which give Euclid "chiefly after the text of Simson." Simson's first 
(Latin) edition (1756) has ''ex versione Latina Federici Commandini" 
on the title-page. Commandinus not only followed the original Greek 
more closely than his predecessors but added to his translation some 
ancient scholia as well as good notes of his own. The title of his 
work is 

Euclidis elementorum libri XV, una cum scholiis antiquis. 
A Federico Comtnandino Urbinate nuper in latinum conversi, 
commentariisque quibusdam iUustrati (Pisauri, apud CamiUum 
Francischinum). I 

He remarks in his preface that Orontius Finaeus had only edited ' 
six Books without reference to any Greek MS., that Peletarius had 1 
followed Campanus' version from the Arabic rather than the Greek 1 
text, and that Candalla had diverged too far from Euclid, having 
rejected as inelegant the proofs given in the Greek text and 1 
substituted faulty proofs of his own. Commandinus appears to have 



1 



Described by Boiicom|Migni, BulkUmo^ II. p. 389. 



I 



CH. viii] TRANSLATIONS AND EDITIONS 105 

used, in addition to the Basel editio princeps^ some Greek MS., so far 
not identified ; he also extracted his *' scholia antiqua " from a MS. 
of the class of Vat. 192 containing the scholia distinguished by 
Heiberg as "Schol. Vat" New editions of Commandinus' translation 
followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and 
Stone), 1756 (Books I. — vi., xi., xil. in Latin and English, by Simson), 
1763 (Keill). Besides these there were many editions of parts of the 
whole work, e.g. the first six Books. 

1574. The first edition of the Latin version by Clavius* 
(Christoph Schlussel, born at Bamberg 1537, died 161 2) appeared 
in 1574, and new editions of it in 1589, 1591, 1603. 1607, 1612. It is 
not a translation, as Clavius himself states in the preface, but it 
contains a vast amount of notes collected from previous commentators 
and editors, as well as some good criticisms and elucidations of his 
own. Among other things, Clavius finally disposed of the error by 
which Euclid had been identified with Euclid of Megara. He speaks 
of the diflferences between Campanus who followed the Arabic 
tradition and the " commentaries 01 Theon," by which he appears to 
mean the Euclidean proofs as handed down by Theon ; he complains 
of predecessors who have either only given the first six Books, or 
have rejected the ancient proofs and substituted worse proofs of their 
own, but makes an exception as regards Commandinus, ''a geometer 
not of the common sort, who has lately restored Euclid, in a Latin 
translation, to his original brilliancy." Clavius, as already stated, did 
not give a translation of the Elements but rewrote the proofs, com- 
pressing them or adding to them, where he thought that he could 
make them clearer. Altogether his book is a most useful work. 

1 62 1. Henry Savile's lectures {Praelectiones tresdecim in prin- 
cipium Elenuntorum Euclidis Oxoniae habitae MDC.XX., Oxonii 162 1), 
though they do not extend beyond I. 8, are valuable because they 
grapple with the difficulties connected with the preliminary matter, 
the definitions etc., and the tacit assumptions contained in the first 
propositions. 

1654. Andr6 Tacquet's Elementa geametriae planae et solidae 
containing apparently the eight geometrical Books arranged for 
general use in schools. It came out in a large number of editions up 
to the end of the eighteenth century. 

1655. Barrow's Euclidis Elementorum Libri XV breviter demon-- 
strati is a book of the same kind. In the preface (to the edition of 
1659) he says that he would not have written it but for the fact that 
Tacquet gave only eight Books of Euclid. He compressed the work 
into a very small compass (less than 400 small pages, in the edition 
of 1659, for the whole of the fifteen Books and the Data) by abbre- 
viating the proofs and using a large quantity of symbols (which, he 
says, are generally Oughtr^'s). There were several editions up to 
1732 (those of 1660 and 1732 and one or two others are in English). 

1 Euclidis eUmmtot-um lUni XV. Accessit xvi. de solidorum reguiarium comparatione, 
Omtus terspicuis demonstraHambuSt accuratisque sektdiis Ulustrati, Aucicre Ckristophoro 
Clamo (Romae, apud Vincentiam Accoltum), 1 vols. 



io6 INTRODUCTION [ch. viii 

1658. Giacomo Alfonso Borelli (1608-1679) published EmMks 
resHtutus^ on apparently similar lines, which went Ummgh time mofe 
editions (one in Italian, 1663). 

1660. Claude Franfois Milliet Dechales' eight geometrical Books 
of Euclid's Elements made easy. Dechales' versions of the Etemmts 
had great vogue, appearing in French, Italian and English as wdl 
as Latin. Riccardi enumerates over twenty editions. 

1733. Sacdieri's EucUdes ab om$$i maeva vimUcatms swe emuOus 
geometricus quo stabiliuntur prima ^sa gmmutrieie frimci^ is 
important for his elaborate attempt to prove the parallel-postulate 
forming an important stage in the history of the demopment of non- 
Euclidean geometry. 

i7S6- Simson's first edition, in Latin and in English. The Latin 
title is 

Euclidis elementorum libri prions Mr, Uim mmd eeim us ei du0- 
decimus, ex versiotie latina Pederid Cammandini; sublaiis Us 
quibus olim Ubri hi a Tkeone^ aUisve, vitiaH sumi, ei quiiusdam 
Euclidis demonstrationibus resHiuiis. A Roberto SAmsom Jf.D. 
Glasguae, in aedibus Academids excuddMUit Robertus et Andreas 
Foulis, Academiae typografdiL 

1802. Euclidis elementorum Ubri prims xn ex Com$nandimi ei 
Gregorii versionibus latinis. In usmn iuventuOs Aeademicae...hy 
Samuel Horsley, Bishop of Rochester. (Oxft>rd, Clarendcm Press.) 

IV. Italian versioms or oomcentaries. 

1543. Tartaglia's version, a second edition of which was pub- 
lished in 1565^ and a third in 1585. It does not appear that he used 
anv Greek text, for in the edition of is6s he mentions as available 
only ''the first translation by Campano," "the second made by 
Bartolomeo Zamberto Veneto who is still alive," "the editions of 
Paris or Germany in which they have included both the aforesaid 
translations," and "our own translation into the vulgar (tongue)." 

1575. Commandinus' translation turned into Italian and revised 
by him. 

161 3. The first six Books "reduced to practice" by Pietro 
Antonio Cataldi, re-issued in 1620, and followed by Books vii. — ix. 
(1621) and Book x. (1625). 

1633. Borelli's Latin translation turned into Italian by Domenico 
Magni. 

1680. Euclide restituto by Vitale Giordano. 

169a Vincenzo Viviani's Elementi piani e solidi di Eudide 
(Book v. in 1674). 

1 The title-ptge of the edition of 1565 is as follows : EuciuU Magurrmu phiUsoptU^ s§U 
imirmbitUrt dMe seimiie imaiMsmtOieit diligmUmsnte tyusifUU^, ei alia inttgritH ridotu^ per U 
degmo prefessore di tal seientie NicoU Teartaka Briseianp, stimdo U due traJottiam, eon una 
empla etpositwme dtlU istaso traddtore di m$iwc aggiwUa, talmenU chiara^ ctu ogm wudiccre 
ingegm*^ stmna la MOtilia, otter sujfragio di oUtnT altra uimtia emt facUiU jm empau a 
poiirU imUmdtn^ In Veneda, Appresio Cmtio Troiano, 1565. 



^ 



CH. viii] TRANSLATIONS AND EDITIONS 107 

173 1. EUnufiti geametrici piani e solidi di Euclide by Guido 
Grandi. No translation, but an abbreviated version, of which new 
editions followed one another up to 1806. 

1749. Italian translation of Dechales with Ozanam*s corrections 
and additions, re-issued 1785, 1797. 

1752. Leonardo Ximenes (the first six Books). Fifth edition, 
1819. 

18 1 8. Vincenzo Flauti's Corso di geotnetria eUmentare e sublime 
(4 vols.) contains (Vol. I.) the first six Books, with additions and a 
dissertation on Postulate 5, and (Vol. II.) Books XL, XII. Flauti 
also published the first six Books in 1827 and the Elements of geometry 
of Euclid in 1843 and 1854. 

V. German. 

1558. The arithmetical Books vil.— ix. by Scheubel* (cf. the 
edition of the first six Books, with enunciations in Greek and Latin, 
mentioned above, under date 1550). 

1562. The version of the first six Books by Wilhelm Holtzmann 
(Xylander)*. This work has its interest as the first edition in German, 
but otherwise it is not of importance. Xylander tells us that it was 
written for practical people such as artists, goldsmiths, builders etc., 
and that, as the simple amateur is of course content to know facts, 
without knowing how to prove them, he has often left out the proofs 
altogether. He has indeed taken the greatest possible liberties with 
Euclid, and has not grappled with any of the theoretical difficulties, 
such as that of the theory of parallels. 

165 1. Heinrich Hoffmann's Teutscher Euclides {2nd edition 1653), 
not a translation. 

1694. Ant Ernst Burkh. v. Pirckenstein's Teutsch Redender 
Euclides (eight geometrical Books), "for generals, engineers etc." 
"proved in a new and quite easy manner." Other editions 1699, 

1744. 

1697. Samuel Reyher's In teutscher Sprache vorgestellter Euclides 
(six Books), "made easy, with symbols algebraical or derived from the 
newest art of solution." 

1 7 14. Euclidis XV Backer teutsch, "treated in a special and 
brief manner, yet completely," by Chr. Schessler (another edition in 
1729). 

1773. The first six Books translated from the Greek for the 
use of schools by J. F. Lorenz. The first attempt to reproduce 
Euclid in German word for word. 

1 78 1. Books XL, XII. by Lorenz (supplementary to the pre- 
ceding). Also EuklicTs Elemente funfzehn Bucher translated from 

^ Das sibendacht und nmnt buck des hochberiimbten Mathematici Euclidis Megarensis... 
durch Magistrum Johann Sckeybl^ der loblichen univtrsiUt tu Tiibingaty des Euclidis und 
Arithmclic Ordinaritn^ auss dem latein ins teutsch gebracht..., 

• Die seeks erste Biicher Euclidis vom an fang oder grundder Geometry... Auss Griechiscker 
sprach in die Teiitsck gebracht aigentlich erkldrt...Demcusen vormals in TeUtseher sprach nie 
gueken warden... Durch Wilhelm HoUtman genani Xylander von Augspterg. Getruckht <a 



io8 INTRODUCTION [cH. vui 

the Greek by Lorenz (second edition 1798 ; editions of 1809^ 1818, 
1824 by MoUweide, of 1840 by Dippe). The edition of 1834, and 
I presume those before it, are shortened by the use of symbols and 
the compression of the enunciation and ^setting-out" into one. 

1807. Books I.— VL, Xl^ XIL ''newly transuited from the Greekp" 
by J. K. F. HaufT. 

1828. The same Books by Joh. Jos. Ign. Hoffmann ''as guide 
to instruction in elementary geometry, followed in 1832 by obmrva- 
tions on the text by the same editor. 

1833. Die GeometrU des BukUd und das Wksm dirsMm by 
E. S. Unger; also 1838, 185 1. 

1901. Max Simon, Ettclid und dU Sichs planimetrischm Bikfur. 

VI. French. 

1564-1566. Nine Books translated by Pierre Forcadd, a pupil 
and friend of P. de la Ramde. 

1604. The first nine Books translated and annotated by Jean 
Errard de Bar-le-Duc; second edition, 1605. 

161 5. Denis Henrion's translation of the 15 Books (seven 
editions up to 1676). 

1639. The first six Books ^demonstrated by symtxds, by a 
method very brief and intelligible^'' by Pierre Hdngone, mentioned 
by Barrow as the only editor who, before him, had lued symbols for 
the exposition of Euclid. 

1672. Eight Books "rendus plus fiidles*' by Claude Francois 
Milliet Dechales, who also brought out Les iUtnens d'Euclide ex- 
pliquis d*un€ maniire nouvelle et trh facile^ which appeared in many . 
editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), I 
and was translated into Italian (1749 etc.) and English (by William ^ 
Halifax, 1685). 1 

1804. In this year, and therefore before his edition of the Greek 
text, F. Peyrard published the Elements literally translated into 
French. A second edition appeared in 1809 ^^^ ^^ addition of the 
fifth Book. As this second edition contains Books I. — vi. XI., XII. 
and X. I, it would appear that the first edition contained Books I. — ^iv., 
VI., XL, XII. Peyrard used for this translation the Oxford Greek text 
and Simson. 

VII. Dutch. 

1606. Jan Pieterszoon Dou (six Books). There were many later 
editions. Kastner, in mentioning one of 1702, says that Dou explains 
in his preface that he used Xylander's translation, but, having after- 
wards obtained the French translation of the six Books by Errard 
de Bar-le-Duc (see above), the proofs in which sometimes pleased 
him more than those of the German edition, he made his Dutch 
version by the help of both. 

16 1 7. Frans van Schooten, "The Propositions of the Books of 
Euclid's Elements'*; the fifteen Books in this version ''enlarged" by 
Jakob van Leest in 1662. 

1695. C. J. Voc^ht, fifteen Books complete, with Candalla's •' 16th.'' 



CH. viiij TRANSLATIONS AND EDITIONS 109 

1702. Hendrik Coets, six Books (also in Latin, 1692); several 
editions up to 1752. Apparently not a translation, but an edition for 
school use. 

1763. Pybo Steenstra, Books I. — Vl., XL, XII., likewise an abbre- 
viated version, several times reissued until 1825. 

1 VIII. English. 

f 1570 saw the first and the most important translation, that of Sir 

Henry Billingsley. The title-page is as follows : 

THE ELEMENTS 

OF GEOMETRIE 

of the most auncient Philosopher 

EVCLIDE 

of Megara 

Faithfully {now first) translated into the Englishe toung^ 
by H. Billingsley, Citizen of London. Whereunto are annexed 
certaine Scholies^ Annotations^ and Inuentions^ of the best 
Mathematidens, both of time pasty and in this our age. 

With a very fruitfull Preface by M. I. Dee, specifying the 
chiefe Mathematicall Scieces, what they are^ and whereunto 
commodious: where^ also, are disclosed certaine new Secrets 
Mathematicall and Mechanically vntill these our daies, greatly 
missed. 

Imprinted at London hy fohn Daye. 

The Preface by the translator, after a sentence observing that with- 
out the diligent study of Euclides Elementes it is impossible to attain 
unto the perfect knowledge of Geometry, proceeds thus. " Wherefore 
considering the want and lacke of such good authors hitherto in our 
Englishe tounge, lamenting also the negligence, and lacke of zeale to 
their countrey in those of our nation, to whom God hath geuen both 
knowledge and also abilitie to translate into our tounge, and to 

Ipublishe abroad such good authors and bookes (the chiefe instrumentes 
of all leaminges): seing moreouer that many good wittes both of 
gentlemen and of others of all degrees, much desirous and studious of 
F these artes, and seeking for them as much as they can, sparing no 
paines, and yet frustrate of their intent, by no meanes attaining to 
that which they seeke : I haue for their sakes, with some charge and 
great trauaile, faithfully translated into our vulgare touge, and set 
abroad in Print, this booke of Euclide. Whereunto I haue added 
easie and plaine declarations and examples by figures, of the defini- 
tions. In which booke also ye shall in due place finde manifolde 
I additions, Scholies, Annotations, and Inuentions: which I haue 
gathered out of many of the most famous and chiefe Mathematicies, 
both of old time, and in our age : as by diligent reading it in course, 
ye shall well perceaue...." 

It is truly a monumental work, consisting of 464 leaves, and there- 
fore 928 pages, of folio size, excluding the lengthy preface by Dee. 
The notes certainly include all the most important that had ever been 



no INTRODUCTION [ck. vui 

written, from those of the Greek commeiitators, Proclus and the others 
whom he quotes, down to those of Dee himself on the last books. 
Besides the fifteen Books, Billingsley included the ''sixteenth" added 
by Candalla. The print and appearance of the book are worthy of its 
contents ; and, in order that it may be understood how no pains were 
spared to represent everything in the dearest and most perfect form, . 
I need only mention that the figures of the propositions in Book XL J 
are nearly all duplicated, one being the figure of Euclid, the other an 1 
arrangement of pieces of paper (triangular, rectang^ular etc) pasted at 
the edges on to the page of the book so that the pieces can be turned 
up and made to show the real form of the solid figures represented. 

Billingsley was admitted Lady Margaret Scholar of St John's 
College, Cambridge, in 1 55 1, and he is also said to have studied at i 
Oxford, but he did not take a d^ree at either University. He was ^ 
afterwards apprenticed to a London haberdasher and rapidly became 
a wealthy merchant Sheriff of London in 1584, he was elected Lord 
Mayor on 3 ist December, 1 596^ on the death, during his year of oflSce^ 
of Sir Thomas Skinner. From 1589 he was one of the Queen's four 
" customers,'* or farmers of customs, of the port of London. In 1591 
he founded three scholarships at St John's Collie for poor students, 
and gave to the College for thdr noAintenance two messuages and 
tenements in Tower Street and in Mark Lane, AllhallowSi Barking. 
He died in 1606. 

165 1. Elements of Geometry, The first VI Boocks: In a eompeti^ 
diousform contracted and demomtraUd by Captain Thomas Rudd, with 
the mathematicall preface of John Dee (London). 

1660. The first English edition of Barrow's Euclid (published in 
Latin in 1655), appeared in London. It contained "the whole fifteen 
books compendiously demonstrated"; several editions followed, in 
1705, 1722, 1732, 1751. 

1661. Eudids Elements of Geometry^ with a supplement of divers 
Propositions and Corollaries, To which is added a Treatise of regular 
Solids by Campane and Flussat ; likewise EuclicTs Data and Marinus 
his Preface. Also a Treatise of the Divisions of Superficies^ ascribed to 
Machomet Bagdedine^ but published by Commandine at the request of 
f. Dee of London. Published by care and industry of John Leeke and 
Geo. Serle, students in the Math. (London). According to Potts this 
was a second edition of Billingsley's translation. 

1685. William Halifax's version of Dechales' " Elements of Euclid i 
explained in a new but most easy method " (London and Oxford). 

1705. The English Euclide; being the first six Elements of I 
Geometry^ translated out of the Greeks with annotations and useful/ ( 
supplements by Edmund Scarburgh (Oxford). A noteworthy and 4 
useful edition. \ 

1708. Books L — ^VL, XL, XII., translated from Commandinus' Latin 
version by Dr John Keill, Savilian Professor of Astronomy at Oxford. 

Keill complains in his preface of the omissions by such editors as 
Tacquet and Dechales of many necessary propositions (e.g. VL 27 — 29), 
and of their substitution of proofs of their own for Euclid's. He praises 
Barrow's version on the whole, though objecting to the ** algebraical " 



CH. viii] TRANSLATIONS AND EDITIONS iix 

form of proof adopted in Book ll., and to the excessive use of notes 
and symbols, which (he considers) make the proofs too short and 
thereby obscure: his edition was therefore intended to hit a proper 
mean between Barrow's excessive brevity and Clavius' prolixity. 

Keiirs translation was revised by Samuel Cunn and several times 
reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782 
the twelfth. 

1 7 14. W. Whiston's English version (abridged) of The Elements 
of Euclid with select theorems out of Archimedes by the learned Andr. 
Tacguet. 

1756. Simson's first English edition appeared in the same year as 
his Latin version under the title : 

The Elements of Euclid, viz, the first six Books together with 
the eleventh and twelfth. In this Edition the Errors by which 
Theon or others have long ago vitiated these Books are corrected mtd 
some of Euclid's Demonstrations are restored. By Robert Simson 
(Glasgow). 

As above stated, the Latin edition, by its title, purports to be "ex 
versione latina Federici Commandini,*' but to the Latin edition, as well 
as to the English editions, are appended 

Notes Critical and Geometrical; containing an Account of those 
things in which this Edition differs from the Greek text; and ike 
Reasons of the Alterations which have been made. As also Obser- 
vations on some of the Propositions. 

Simson says in the Preface to some editions (e.g. the tenth, of 
1799) ^^^^ '*^^ translation is much amended by the friendly assistance 
of a learned gentleman." 

Simson's version and his notes are so well known as not to need 
any further description. The book went through some thirty suc- 
cessive editions. The first five appear to have been dated I7S6, 1762, 
1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806, 
the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844. 
The DcUa '' in like manner corrected " was added for the first time in 
the edition of 1762 (the first octavo edition). 

1 78 1, 1788. In these years respectively appeared the two volumes 
containing the complete translation of the whole thirteen Books by 
James Williamson, the last English translation which reproduced 
Euclid word for word. The title is 

The Elements of Euclid, with Dissertations intended to assist 
and encourage a critical examination of these Elements, as the most 
effectual means of establishing a juster taste upon mathematical 
} subjects than that which at present prevails. By James Williamson. 

tin tfie first volume (Oxford, 1781) he is described as "MA. 
Fellow of Hertford College," and in the second (London, printed by 
T. Spilsbury, 1788) as "B.D." simply. Books v., vi. with the Con- 
) elusion in the first volume are paged separately from the rest 

1 78 1. An examination of the first six Books of Euclid's Elements, 
by William Austin (London). 

1795- John Playfair's first edition, containing "the first six Books 
of Euclid with two Books on the Geometry of Solids." The book 



INTRODUCTION [CB. Tin 

a fifth edition in 1819, an eighth in i83l,a ninth in 1836^ and 
Ith in 1846. 

[826. Riccardi notes under this* date Euclid's EUmmis 9f Gta- 

* containing the whole twelve Books translmtidinio Et^gKsktfr^m ike 

ofPeyrard^ by George Phillips. The editor, who waft rasident 

FQueens' College, Cambridge, 1857-1892, was bom in 1804 and 

jltriculated at Queens' in 1826, so that he must have published the 

3k as an undergraduate. 

1828. A very valuable edition of the fiist six Books is diat of 
l^ionysius Lardner, with commentary and ge om etrical exerdses, to 
Irhich he added, in place of Books XL, XIL, a Treatise on Solid 
reometry mostly based on L^endre. Lardner compresses the pro- 
sitions by combining the enunciation and tihe setting-out, auid lie 
[gives a vast number of riders and additional propositions in smaller 
print The book had reached a ninth edition by 184^ and an eleventh 
by 185s. Among other things, Lardner gives an Appendix ^on the 
theory of parallel lines," in which he gives a short histoiy of the 
attempts to get over the difficulty of the parallel-postulate^ down to 
that of Legendre. 

1833. T. Perronet Thompson's Gmmutvy wiUumt axioms^ or ike 
first Book of EuclicTs Elements wiik aUerwOons emd fMes; emd em 
intercalary book in which the straight Urn emd fleme are dnkfod from 
properties of tlu sphere ^ with an appendix coniamit%g noOces of m e ikods 
proposed for getting over the diffictUty in ike iwe^k euHam ofEneUd 

Thompson (1783-1869) was 7th wrangler 1802, mklshipman 1803, 
Fellow of Queens' College, Cambridge, 1804, and afterwards general 
and politician. The book went through several editions, but» naving 
been well translated into French by Van Tenac» is nid to have 
received more recognition in France than at home. 

1845. Robert Potts' first edition <(and one of the best) entitled: 

Euclid's Elements of Geometry chiefly from the text of 
Dr Simson with explanatory notes.. Jo which is prefixed an 
introduction containing a brief outline of the History of Geometry. 
Designed for tlu use of the higher forms in Public Schools a»id 
students in the Universities (Cambridge University Press, and 
London, John W. Parker), to which was added (1847) ^^ 
Appendix to the larger edition of Euclid s Elements of Geometry^ 
containing additional notes on the Elements, a short tract on trans- 
versalSy and hints for the solution of the problems etc. 
1862. Todhunter's edition. 

The later English editions I will not attempt to enumerate ; their 
name is l^ion and their object mostiy that of adapting Euclid for school 
use, with all possible gradations of departure from his text and order. 

IX. Spanish. 

1576. The first six Books translated into Spanish by Rodrigo 
^amorano. 

1637. The first six Books translated, with notes, by L. Carduchi. 

1689. Books L — VL, XL, xiL, translated and explained by Jacob 
Knesa. 



1 



CH. viii] TRANSLATIONS AND EDITIONS 113 

X. Russian. 

1739. Ivan Astaroff (translation from Latin). 

1789. Pr. SuvorofTand Yos. Nikitin (translation from Greek). 

1 880. Vachtchenko-Zakhartchenko. 

(18 1 7. A translation into Polish by Jo. Czecha.) 

XI. Swedish. 

1744. Mirten Stromer, the first six Books; second edition 1748. 
The third edition (1753) contained Books XI. — xii. as well; new 
editions continued to appear till 1884. 

1836. H. Falk, the first six Books. 

1844, J 84s, 1859. P. R. BrSkenhjelm, Books I. — VI., XI., XII. 

i8sa F. A. A. Lundgren. 

1850. H. A. Witt and M. E. Areskong, Books I.— vi., xi., xil. 

XII. Danish. 

174s. Ernest Gottlieb Ziegenbalg. 
1803. H. C. Linderup, Books I. — VI. 

XIII. Modern Greek. 
1820. Benjamin of Lesbos. 






H. E. 



-*■ -i^- 



-t.V* ,Hr > 






CHAPTER IX. 

{ 1, ON THE NATURE OF ELEMENTS. 

It would not: be easy to find a more lucid explanation of the terms 
tknunt and eUmentafy, and of the distinction between them, than 
is found in Proc1us\ who is doubtless, here as so often, quoting 
from Geminus, There are, says Proclus, in the whole of geometry 
certain leading theorems, bearing to those which follow the relation of I 
a principle, all- pervading, and furnishing proofs of many properties* 
Such theorems are called by the name of tlemintsx and their function 
may be compared to that of the letters of the alphabet in relation to 
language, letters being indeed called by the same name in Greek 

The term elententaty, on the other hand, has a wider application : 
it is applicable to things "which extend to greater multiplicity, and, 
though possessing simplicity and elegance, have no longer the same 
dignity as the eUments^ because their investigation is not of general 
use in the whole of the science, eg, the proposition that in triangles 
the perpendiculars from the angles to the trans vei^e sides meet in a 
point." 

" Again, the term element is used in two senses, as Menaechmus 
says. For that which is the means of obtaining is an element of that 
which is obtained, as the first proposition in Euclid is of the second, 
and the fourth of the fifth. In this sense many things may even be 
said to be elements of each other, for they are obtained from one 
anotiier. Thus from the fact that the exterior angles of rectilineal 
figures are (together) equal to four right angles we deduce the number 
of right angles equal to the internal angles (taken together)*, and 
via versa. Such an element is like a lemma. But the term element is 
otherwise used of that into which, being more simple, the composite is 
divided ; and in this sense we can no longer say that everything is an 
element of everything, but only that things which are more of the ^ 
nature of principles are elements of those which stand to them In the 
relation of results, as postulates are elements of theorems. It is 

^ Prodos, Comm, on Etui, i., ed. Friedleio, f>p. 7a sqq. 

* T^ vX^of rfir hrrht 6p0tut f^wr. If the text is nght, we most apparently take it as ''the 
number of the angles equal to right angles that there are inside,'* i.e. that are made up by 
the internal angles. 



I 



CH. IX.J1] ON THE NATURE OF ELEMENTS 115 

according to this signification of the term element that the elements 
found in Euclid were compiled, being partly those of plane geometry, 
and partly those of stereometry. In like manner many writers have 
drawn up elementary treatises in arithmetic and astronomy. 

*' Now it is difficult, in each science, both to select and arrange in 
due order the elements from which all the rest proceeds, and into 
which all the rest is resolved. And of those who have made the 
attempt some were able to put together more and some less ; some 
used shorter proofs, some extended their investigation to an indefinite 
length ; some avoided the method of reductio ad absurdum^ some 
2l\o\A^ proportion \ some contrived preliminary steps directed against 
those who reject the principles ; and, in a word, many different 
methods have been invented by various writers of elements. 

" It is essential that such a treatise should be rid of everything 
superfluous (for this is an obstacle to the acquisition of knowledge) ; 
it should select everything that embraces the subject and brings it to 
a point (for this is of supreme service to science) ; it must have great 
r^ard at once to clearness and conciseness (for their opposites trouble 
our understanding); it must aim at the embracing of theorems in 
general terms (for the piecemeal division of instruction into the more 
partial makes knowledge difficult to grasp). In all these ways 
Euclid's system of elements will be found to be superior to the rest ; 
for its utility avails towards the investigation of the primordial 
figures*, its clearness and organic perfection are secured by the 
precession from the more simple to the more complex and by the 
foundation of the investigation upon common notions, while generality 
of demonstration is secured by die progression through the theorem's 
which are primary and of the nature of principles to the things sought. 
! As for the things which seem to be wanting, they are partly to be 
1 discovered by the same methods, like the construction of the scalene 
• and isosceles (triangle), partly alien to the character of a selection of 
elements as introducing hopeless and boundless complexity, like the 
! subject of unordered irrationals which Apolionius worked out at 
length', and partly developed from things handed down (in the 
elements) as causes, like the many species of angles and of lines. 
These things then have been omitted in Euclid, though they have 
received full discussion in other works ; but the knowledge of them is 
derived from the simple (elements)." 

Proclus, speaking apparently on his own behalf, in another place 
distinguishes two objects aimed at in Euclid's Elements. ZTEe first 
has reference to the matter of the investigation, and here, li^e a good 
Platonist, he takes the whole subject of geometry to be concerned 
with the " cosmic figures," the five regular solids, which in Book XIIL 

* r(a9 d^x^'f^ ^iiM^rciir, by which Proclus probably means the regular polyhedxa 
(Tannery, p. i43»-)- 

' We have no more than the most obscure indications of the character of this work in an 

. Arabic MS. analysed by Woepcke, Essed d^une restitution de travaux perdm dAtolUmius 

tur Us fuantith irrationelUs d*apris da indicaticHS tirits d*un manuscrit arabi in Mhnoires 

fristntis h ttuadhnie da sciences^ XI v. 658—720, Paris, 1856. Cf. Cantor, Gtsch. d, Maik. 

't* PP* 34^~~9 * details are also given in my notes to Book x. 

i' 8-a 



li 



1 16 INTRODUCTION [CH. dl f i 

are constructed, inscribed in a sphere and compared with one another. 
The second object is relative to the learner; and, from this standpoint, 

^the elements may be described as ''a means of perfecting the learnei's 
understanding with reference to the whole of geometiy. | For, starting 
from these (elements), we shall be able to acquire knowledge of the 
other parts of this science as well, while without them it is impossible 
for us to get a grasp of so complex a subject, and knowledge ci the 
rest is unattainable. As it is, the theorems which are most of the 
nature of principles, most simple, and most akin to the first hypotheses 
are here collected, in their appropriate order; and the proon of all 
other propositions use these theorems as thoroughly well known, and 
start from them. Thus Archimedes in the books on the sphere and 
cylinder, Apollonius, and all other geometers, clearly use the theorems 

proved in this very treatise as constituting admitted principles\'' 

Aristotle too speaks of elements of geometry in the same sense. 
Thus: "in geometry it is well to be thoroughly versed in the 
elements*"; "in general the first of the elements are, given the 
definitions, e.g. of a straight line and of a circle^ most easy to prove, 
although of course there are not many data that can be used to 
establish each of them because there are not many middle terms*"; 
"among geometrical propositions we call those 'elements' the proofs of 
which are contained in the proofs of all or most of such propositions^*'; 
"(as in the case of bodies)^ so in like manner we speak of the elements 

' of geometrical propositions and, generally, of demonstrations ; for the 
demonstrations which come first and are attained in a variety of 
other demonstrations are called elements of those demonstrations... 
the term element is applied by analogy to that which, being one and 
small, is useful for many purposes'." 



§ 2. ELEMENTS ANTERIOR TO EUCLID'S. 

The early part of the famous summary of Proclus was no doubt 
drawn, at least indirectly, from the history of geometry by Eudemus ; 
this is generally inferred from the remark, made just after the mention 
of Philippus of Mende, a disciple of Plato, that "those who have 
written histories bring the development of this science up to this 
point" We have therefore the best authority for the list of writers of 
elements given in the summary. Hippocrates of Chios (fl. in second 
half of 5th c.) is the first ; then Leon, who also discovered diarismiy 
put together a more careful collection, the propositions proved in it 
being more numerous as well as more serviceable*. Leon was a little 
older than Eudoxus (about 390-337 B.C) and a little younger than 
Plato (429-348 B.C.), but did not belong to the latter's school. The 

* Prodtts, pp. 70, 10 — 71, ai. 

* T^s VIII. 14, 163 b as. » To^s viii. 3, 158 b 35. * Mett^k. 998 a as. 

* ifaath, loi^a 35— b 5. 

* Prodos, p. 66, ao iSrrt rdr Atforra iral tA 9roix«<a «wtft4Mu r^ rt rX^^i k9X rj %pd^ 
rQf9 dmnnf/thttif iwifUkkm^. 



I 



CH. ix.§a] ELEMENTS ANTERIOR TO EUCLID'S 117 

geometrical text-book of the Academy was written by Theudius of 
Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of 
Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus 
consorted together in the Academy and carried on their investigations 
in common. Theudius " put together the elements admirably, making 
many partial (or limited) propositions more generals" Eudemus 
mentions no text-book after that of Theudius, only adding that Her- 
motimus of Colophon "discovered many of the elements"." Theudius 
then must be taken to be the immediate precursor of Euclid, and no 
doubt Euclid made full use of Theudius as well as of the discoveries of 
Hermotimus and all other available material. Naturally it is not in 
Euclid's Elements that we can find much light upon the state of the 
subject when he took it up ; but we have another source of informa- 
tion in Aristotle. Fortunately for the historian of mathematics, 
Aristotle was fond of mathematical illustrations ; he refers to a con- 
siderable number of geometrical propositions, definitions etc., in a 
way which shows that his pupils must have had at hand some text- 
book where they could find the things he mentions; and this text-book 
must have been that of Theudius. Heiberg has made a most valuable 
collection of mathematical extracts from Aristotle*, from which much 
is to be gathered as to the changes which Euclid made in the methods 
of his predecessors ; and these passages, as well as others not included 
in Heiberg's selection, will often be referred to in the sequel. 

§3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, 
AND AXIOMS. 

On no part of the subject does Aristotle give more valuable 
information than on that of the first principles as, doubtless, generally 
accepted at the time when he wrote. One long passage in the 
Posterior Analytics is particularly full and lucid, and is worth quoting 
in extenso. After laying it down that every demonstrative science 
starts from necessary principles^ he proceeds': 

^ By first principles in each genus I mean those the truth of which 
It is not possible to prove. What is denoted by the first (terms) and 
those derived from them is assumed ; but, as regards their existence, 
this must be assumed for the principles but proved for the rest. Thus 
what a unit is, what the straight (line) is, or what a triangle is (must 
be assumed) ; and the existence of the unit and of magnitude must 
also be assumed, but the rest must be proved. Now of Uie premisses 
used in demonstrative sciences some are peculiar to each science and 
others common (to all), the latter being common by analogy, for of 
course they are actually useful in so far as they are applied to the sub- 
ject-matter included under the particular science. Instances of first 

^ Proclus, P' 67, 14 icoi 7dp rd rroixeSa Ka\(at avpira^tp koI roXKii rOif fitpucwp [6piKQp (?) 
Friedlein] KaBoXucirtpa iwolyfatw. 

' Proclus, p. 67, 33 Twv CTOixtif^ roXXd dptvp€. 

* Maikemaiiscka mu AristoUies in Abhandlungen wur Gtsch, d, math. Wissenichafttn^ 
xvni. Heft (1904), pp. I— 49* 

^ Anal, post, I. 6, 74 b 5. * ibitL i. 10, 76 a 31 — 77 a 4. 



1 18 INTRODUCTION [ch. ul f 3 

principles peculiar to a science are fhe assumptions that a Une is of 
such and such a character, and similarly for the straight (line); whereas 
it is a common principle, for instance, that, if equals be subtracted 
from equals, the remainders are equal. But it is enough that each tA 
the common principles is true so far as regards the particular genus 
(subject-matter) ; for (in geometiy) the effect will be the same even if 
the common principle be assumed to be true, not of everything, but 
only of magnitudes, and, in arithmetic, of numbers. 

'* Now the things peculiar to the science, the existence of which 
must be assumed, are the things with r efer en ce to which the science 
investigates the essential attributes, e.g. arithmetic with reference to 
units, and geometry with reference to points and lines. With these 
things it is assumed that they exist and that they are of such and 
such a nature. But, with regard to their essential properties, what is 
assumed is only the meaning of each term employed : thus arithmetic 
assumes the answer to die question what is (meant by) 'odd* or 
'even,' 'a square' or 'a cube,' and geometry to the question 
what is (meant by) 'the irrational ' or 'deflection' or (the so-called) 
'verging' (to a point); but that there are such things is proved by 
means of the common principles and of what has already been 
demonstrated. Similarly with astronomy. For every demonstrative 
science has to do with three things, (1) tne things wUch are assumed 
to exist, namely the genus (subject-matter) in ei^ case, the essential 
properties of which the science investigates, (2) the common axioms 
so-called, which are the prinuuy source of demonstration, and (3) the 
properties with regard to which all that is assumed is the meaning of 
the respective terms used. There is, however, no reason why some 
sciences should not omit to speak of one or other of these things. 
Thus there need not be any supposition as to the existence of the 
genus, if it is manifest that it exists (for it is not equally clear that 
number exists and that cold and hot exist) ; and, with regard to the 
properties, there need be no assumption as to the meaning of terms if 
it is clear : just as in the common (axioms) there is no assumption as 
to what is the meaning of subtracting equads from equals, because it is 
well known. But none the less is it true that there are three things 
naturally distinct, the subject-matter of the proof, the things provod, 
and the (axioms) from which (the proof starts). 

•'Now that which is per se necessarily true, and must necessarily be 
thought so, is not a hypothesis nor yet a postulate. For demon- 
stration has not to do with reasoning from outside but with the 
reason dwelling in the soul, just as is the case with the syllogism. 
It is always possible to raise objection to reasoning from outside, 
but to contradict the reason within us is not always possible. New 
anything that the teacher assumes, though it is matter of proft, 
without proving it himself, is a hypothesis if the thing assumed is 
believed by the learner, and it is moreover a hypothesis, not abso- 
lutely, but relatively to the particular pupil ; but, if the same thing 
b assumed when the learner either has no opinion on the subject 
or is of a contrary opinion, it is a postulate. This is the difference 



CH. DL§3] FIRST PRINCIPLES 119 

between a hypothesis and a postulate ; for a postulate is that which 

is rather contrary than otherwise to the opinion of the learner, or 

I whatever is assumed and used without being proved, although matter 

Ffor demonstration. Now definitions are not hypotheses, for they do 
not assert the existence or non-existence of anything, while hypotheses 
are among propositions. Definitions only require to be understood : 
a definition is therefore not a hypothesis, unless indeed it be asserted 
f that any audible speech is a hypothesis. A hypothesis is that from 
I the truth of which, if assumed, a conclusion can be established. Nor 
i are the geometer's hypotheses false, as some have said : I mean those 
^ who say that ' you should not make use of what is false, and yet the 
I geometer falsely calls the line which he has drawn a foot long when 
t it is not, or straight when it is not straight' The geometer bases no 
' conclusion on the particular line which he has drawn being that which 
he has described, but (he refers to) what is illustrated by the figures. 
Further, the postulate and every hypothesis are either universal or 
particular statements; definitions are neither" (because the subject 
is of equal extent with what is predicated of it). 

Every demonstrative science, says Aristotle, must start from in- 
demonstrable principles : otherwise, the steps of demonstration would 
be endless. Of these indemonstrable principles some are (a) common 
to all sciences, others are {b) particular, or peculiar to the particular 
science ; {a) the common principles are the axioms, most commonly 
illustrated by the axiom that, if equals be subtracted from equals, the 
remainders are equal. Coming now to (b) the principles peculiar to 
the particular science which must be assumed, we have first the genus 
or subject-matter, the existence of which must be assumed, viz. magni- 
tude in the case of geometry, the unit in the case of arithmetic. Under 
this we must assume definitions of manifestations or attributes of the 
genus, e.g. straight lines, triangles, deflection etc. The definition in 
itself says nothing as to the existence of the thing defined : it only 
requires to be understood. But in geometry, in addition to the genus 
and the definitions, we have to assume the existence of a kvf frimary 
things which are defined, viz. points and lines only : the existence 
of everything else, e.g. the various figures made up of these, as 
triangles, squares, tangents, and their properties, e.g. incommensur- 
ability etc., has to be proved (as it is proved by construction and 
demonstration). In arithmetic we assume the existence of the unit: 
but, as regards the rest, only the definitions, e.g. those of odd, even, 
square, cube, are assumed, and existence has to he proved. We have then 
clearly distinguished, among the indemonstrable principles, axioms 
and definitions, A postulate is also distinguished from a hypothesis, 
the latter being made with the assent of the learner, the former 
iRthout such assent or even in opposition to his opinion (though, 
strangely enough, immediately after saying this, Aristotle gives a 
wider meaning to "postulate" which would cover "hypothesis" as well, 
namely whatever is assumed, though it is matter for proof, and used 
without being proved). Heiberg remarks that there is no trace in 
Aristotle of Euclid's Postulates, and that " postulate" in Aristotle has 



X 20 INTRODUCTION [ch. xx. f 3 

a different meaning. He seems to base this on the alternative 
description of postulate, indistinguishable from a hypothesis; but. 
if we take the other description in which it is distinguished fiom a 
hypothesis as being an assumption of something which is a proper 
subject of demonstration without the assent or against the opinion of 
the learner, it seems to fit Euclid's Postulates fainy well, not only the 
first three (postulating three constructionsX but eminently also the other 
two, that all right angles are equal, and that two straight lines meeting 
a third and making the internal angles on the same side of it less than 
two right angles will meet on that side. Aristotle's description also 
seems to me to suit the ''postulates" mth which Archimedes b^ins 
his book On the equilibrium ofplams, namely that equal weights baliuice 
at equal distances, and that equal weights at unequal distances do not 
balance but that the weight at the longer dbtance will prevail 

Aristotle's distinction also between kfpothisis and definiiiaH^ and 
between hypothesis and eua&m^ is clear from the foUowine passa^: 
''Among immediate syllogistic principles, I call that a msis which 
it is neither possible to prove nor essential for any one to hold who 
is to learn anything ; but that which it is necessary for any one to 
hold who is to learn anything whatever is an axiom : for there are 
some principles of this kind, and that is the most usual name 1^ 
which we speak of them. But, of theses, one kind is that which 
assumes one or other side of a predication, as, for instance^ that 
something exists or does not exist, and this is a Ig^thisis ; the other, 
which makes no such assumption, is a defimiioH. For a definition is 
a thesis : thus the arithmetician posits (rlOrrai) that a unit is that 
which is indivisible in respect of quantity ; but this is not a hypo- 
thesis, since what is meant by a unit and the fact that a unit exists 
are different things*." 

Aristotle uses as an alternative term for axioms ''common (things)," 
ra tcoivd, or "common opinions" (tcoipol So^ai), as in the following 
passages. *' That, when equals are taken from equals, the remainders 
are equal is (a) common (principle) in the case of all quantities, but 
mathematics takes a separate department (diroXafiovaa) and directs its 
investigation to some portion of its proper subject-matter, as e.g. lines 
or angles, numbers, or any of the other quantities"." "The common 
(principles), e.g. that one of two contradictories must be true, that 
equals taken from equals etc, and the like*...." " Withr^[ard to the 
principles of demonstration, it is questionable whether they belong to 
one science or to several. By principles of demonstration I mean the 
common opinions from which all demonstration proceeds, e.g. that one 
of two contradictories must be true, and that it is impossible for the 
same thing to be and not be^" Similarly "every demonstrative 
(science) investigates, with r^ard to some subject-matter, the essential 
attributes, starting from the common opinions*** We have then here, 
as Heiberg says, a sufficient explanation of Euclid's term for axioms, 

* Anai, post. I. a, 7a a 14—94. * Mttaph, 1061 b 19—34. 

* Anai. past, 1. 11, 77 a 30. « Mtuipk. 996b 36—30. 

* Mdapk. 997 a so— ^3. 



CH. ix.§3] FIRST PRINCIPLES lai 

viz. common notions {koivoX hpouti), and there is no reason to suppose 
it to be a substitution for the original term due to the Stoics : cf. 
Proclus* remark that, according to Aristotle and the geometers, axiom 
and common notion are the same things 

Aristotle discusses the indemonstrable character of the axioms 
in the Metaphysics. Since "all the demonstrative sciences use the 
axiomsV' the question arises, to what science does their discussion 
belong'? The answer is that, like that of Being {ovala), it is the 
province of the (first) philosopher^ It is impossible that there should 
be demonstration of everything, as there would be an infinite series of 
demonstrations : if the axioms were the subject of a demonstrative 
science, there would have to be here too, as in other demonstrative 
sciences, a subject-genus, its attributes and corresponding axioms* ; thus 
there would be axioms behind axioms, and so on continually. The 
axiom is the most firmly established of all principles*. It is ignorance 
alone that could lead any one to try to prove the axioms' ; the supposed 
proof would be a petitio principii*. If it is admitted that not every- 
thing can be proved, no one can point to any principle more truly 
indemonstrable*. If any one thought he could prove them, he could 
at once be refuted ; if he did not attempt to say anything, it would 
be ridiculous to argue with him : he would be no better than a 
vegetable". The first condition of the possibility of any argument 
whatever is that words should signify something both to the speaker 
and to the hearer: without this there can be no reasoning with any one. 
And, if any one admits that words can mean anything to both hearer 
and speaker, he admits that something can be true without demon- 
stration. And so on". 

It was necessary to give some sketch of Aristotle's view of the 
first principles, if only in connexion with Proclus' account, which is 
as follows. As in the case of other sciences, so ''the compiler of 
elements in geometry must give separately the principles of the 
science, and after that the conclusions from those principles, not 
giving any account of the principles but only of their consequences. 
No science proves its own principles, or even discourses about them : 
they are treated as self-evident... Thus the first essential was to dis- 
tinguish the principles from their consequences. Euclid carries out 
this plan practically in every book and, as a preliminary to the whole 
enquiry, sets out the common principles of this science. Then he 
divides the common principles themselves into hypotheses^ postulates^ 
and axioms. For all these are different from one another : an axiom, 
a postulate and a hypothesis are not the same thing, as the inspired 
Aristotle somewhere says. But, whenever that which is assumed and 
ranked as a principle is both known to the learner and convincing in 
itself, such a thing is an axiom^ e.g. the statement that things which 
are equal to the same thing are also equal to one another. When, on 

^ Proclus, p. 104, 8. * Mdaph. 907 a 10. 

' ibid, 99(5 36. ^ ibid, 1005 a 31— b 11. * ibid. 997 a 5 — 8. 

* ilnd. 1005 b II — 17. ' ibid. 1006 a 5. * ibid. 1006 a 17. 

* fA«/. 1006a la ^ f!^f^ 1006a II— 15. " ftW. 1006 a 18 sqq. 



&. 



123 INTRODUCTION [cB. n. f 3 

the other hand, the pupil has not the notion of what is told him 
which carries conviction in itself, but nevertheless iBys it down and 
assents to its being assumed, such an assumption is a Ig^thnis, 
Thus we do not preconceive by virtue of a common notion, and 
without being taught, that the circle is such and such a figure, but, 
when we are told so, we assent without demonstration. When again 
what is asserted is both unknown and assumed even without the 
assent of the learner, then, he says, we call this a postidaU^ e.g. that 
all right angles are equal This view of a postulate is clearly implied 
by those who have made a spedal and systematic attempt to show, 
with r^[ard to one of the postulates, that it cannot be assented to by 
any one straight off. According then to the teaching of Aristotle, an 
axiom, a postulate and a hypothesis are thus distinguished^** 

We observe, first, that Proclus in this passage confuses kypotlksis 
and definitions, although Aristotle had made the distinction quite 
plain. The confusion may be due to his having in his mind a passage 
of Plato from which he evidently got the phrase about ^ not giving 
an account of" the principles. The passage is*: ^ I think you know 
that those who treat of geometries and calculations (arithmetic) and 
such things take for granted (Airotf^/Myoi) odd and even, figures, 
angles of three kinds, and other things akin to these in each subject, 
implying that they know these things, and, though using them as 
hypotheses, do not even condescend to give any account of them 
either to themselves or to others, but b^n from these things and 
then go through everything else in order, arriving ultimatdy, l^ 
recognised methods, at the conclusion which they started in search 
of." But the hypothesis is here the assumption, e.g. * that there may 
be suck a thing as length without breadth, henceforward called a line*,' 
and so on, without any attempt to show that there is such a thing ; 
it is mentioned in connexion with the distinction between Plato's 
'superior' and 'inferior' intellectual method, the former of which 
uses successive hypotheses as stepping-stones by which it mounts 
upwards to the idea of Good. 

We pass now to Proclus' account of the difference between postu- 
lates and axioms. He begins with the view of Geminus, according 
to which " they differ from one another in the same way as theorems 
are also distinguished from problems. For, as in theorems we propose 
to see and determine what follows on the premisses, while in problems 
we are told to find and do something, in like manner in the axioms 
such things' are assumed as are manifest of themselves and easily 
apprehended by our untaught notions, while in the postulates we 
. assume such things as are easy to find and effect (our understanding 
suffering no strain in their assumption), and we require no complication 
of macluneryV'..."Both must have the characteristic of being simple 

* Proclns, pp. 75, 10—77, *• 

* RipubHc, VI. CIO c. CJf. Aristotle, Nic. Eth^ 1 151 a 17. 

* H. }%c\aKmtjaurHal of Phiiohgy, vol. X. p. 144. 

^ Produs, pp. 178, 13—179, 8. In Ulnstntion Prodos contrasts the drawins of a straight 
line or a circle with the drawing of a *' single-torn spiral " or of an equilateral triangle, the 



i 



*" 



CH. IX. §3] FIRST PRINCIPLES 123 

and readily grasped, I mean both the postulate and the axiom ; but 
the postulate bids us contrive and find some subject-matter (v\v) to 
exhibit a property simple and easily grasped, while the axiom bids us 
assert some essential attribute which is self-evident to the learner, 
just as is the fact that fire is hot, or any of the most obvious things ^" 

Again, says Proclus, '' some claim that all these things are alike 
postulates, in the same way as some maintain that all things that are 
sought are problems. For Archimedes begins his first book on /ft- 
equilibrium^ with the remark ' I postulate that equal weights at equal 
distances are in equilibrium,' though one would rather call this an 
axiom. Others call them all axioms in the same way as some regard 
as theorems everything that requires demonstration'." 

" Others again will say that postulates are peculiar to geometrical 
subject-matter, while axioms are common to all investigation which 
is concerned with quantity and magnitude. Thus it is the geometer 
who knows that all right angles are equal and how to produce in 
a straight line any limited straight line, whereas it is a common notion 
that things which are equal to the same thing are also equal to one 
another, and it is employed by the arithmetician and any scientific 
person who adapts the general statement to his own subjects" 

The third view of the distinction between a postulate and an axiom 
is that of Aristotle above described*. 

The difficulties in the way of reconciling Euclid's classification 
of postulates and axioms with any one of the three alternative views 
are next dwelt upon. If we accept the first view according to which 
an axiom has reference to something known, and a postulate to 
something done, then the 4th postulate (that all right angles are 
equal) is not a postulate ; neither is the 5th which states that, if a 
straight line falling on two straight lines makes the interior angles 
on the same side less than two right angles, the straight lines, if 
produced indefinitely, will meet on that side on which are the angles 
I less than two right angles. On the second view, the assumption that 
two straight lines cannot enclose a space, "which even now," says 
Proclus, "some add as an axiom," and which is peculiar to the 
subject-matter of geometry, like the fact that all right angles are 
equal, is not an axiom. According to the third (Aristotelian) view, 
"everything which is confirmed {irurrovraC) by a sort of demonstration 

spiral requiring more complex machinery and even the equilateral triangle needing a certain 
method. ** For the geometrical intelligence will say that by conceiving a straight line fixed 
at one end but, as regards the other end, moving round the fixed end, and^a point moving 
along the straight line from the fixed end, I have described the single-turn spiral ; for the 
end of the straight line describing a circle, and the point moving on the straignt Une simul- 
taneously, when they arrive and meet at the same point, complete such a spiral. And again, 
if I draw equal circles, join their common point to the centres of the circles and draw a 

I straight line from one of the centres to the other, I shall have the equilateral triangle. 

. These thinsfs then are far from being completed by means of a single act or of a moment's 

' thought" (p. 180, 8— ai). 

, ' Proclus, p. 181, 4— II. 

* It is necessary to coin a word to render dycovpportwr, which is moreover in the plural. 
The title of the treatise as we have it is Equilibria of planes or centres tf gravity of planes in 
Book I and Equilibria of plana in Book ii. 

» Proclus, p. 181, 16---13. * ibid. p. 183, 6—14. • Pp. 118, 119. 



134 INTRODUCnOli fiaiL»t3 

will be a postulate, and whaA h incapable of proof wUi be an aacicMii^* 
This last statement of Produs is h>o8e, as regards the axdom, bacioie 
it omits Aristotle's requirement that the axiom dioold be a iitf- 
evident truth, and one that must be admitted by any one who la to 
learn anything at all, and, as rq^ards the postulate, because Afistade 
calls a postulate something assumed without proof Ihou^i it is 
"matter of demonstration" ((kweUtierim Jh% but says nothing ofim 
^t^ji-demonstration of the postulates. CHi the whme I think it is 
from Aristotle that we get the best idea of what Eudid undtnUmd 
by a postulate and an axiom or common nolioii. Thus ArisMttfs 
account of an axiom as ai prindpie common to all scienoesi wUdi b 
self-evident, though incapaUe of proo^ agrees suffidentiy with tfie 
contents of Euclid's commom naiwms as reduced to five te tiie most 
recent text (not omitting the fourtii, that ^things whidk oolndd^ are I 
equal to one another"). As rc^rds the pasimlaies^ it must be borne f 
in mind that Aristotle says elsewhere* that, ''other things beinff eqQal» 
that proof is the better whidi proceeds from the fewer postulates or 
hypotheses or propositions.** If then we say that a geometer must 
lay down as principles, first certain axicMns or common notions^ and 
then an irreducible mimm$tm of postulates In tfie Aristolelkui saa«e 
concerned only with the subject-matter of geometry, we are n6t hr 
from describing what Euclid in fiurt does. Ais regards the pos^datM 
we may imagine him sayii^: ^ Besides the common nottons there are 
a few other things whidi 1 must a^ume without proof, but idiieh 
differ from the common notioins in that th^ are not sdf«evident 
The learner may or may not be disposed to agree to them ; but he 
must accept them at the outset on the superior authority of his 
teacher, and must be left to convince himself of their truth in- the 
course of the investigation which follows. In the first place certain 
simple constructions, the drawing and producing of a straight line, 
and the drawing of a circle, must be assumed to be possible, and with 
the constructions the existence of such things as straight lines and 
circles ; and besides this we must lay down some postulate to form 
the basis of the theory of parallels." It is true that the admission of 
the 4th postulate that all right angles are equal still presents a 
difficulty to which we shall have to recur. 

There is of course no foundation for the idea, which has found 
its way into many text-books, that *' the object of the postulates is to 
declare that the only instruments the use of which is permitted in 
geometry are the ruU and compassK" 



§ 4. THEOREMS AND PROBLEMS. 

"Again the deductions from the first principles," says Proclus, . 
"are divided into problems and theorems, the former embracing the ( 



* Proclus, pp. 18a, 91 — 183, 13. 

' Cf. Lardner's Endid : also Todhnnter. 



• Atta/. poit. I. a5, 86 a 33—35. 



CH. ix.$4] THEOREMS AND PROBLEMS 125 

generation, division, subtraction or addition of figures, and generally 
the changes which are brought about in them, the latter exhibiting 
the essential attributes of each*." 

" Now, of the ancients, some, like Speusippus and Amphinomus, 
thought proper to call them all theorems, regarding the name of 
theorems as more appropriate than that of problems to theoretic 
sciences, especially as these deal with eternal objects. For there is 
no becoming in things eternal, so that neither could the problem 
have any place with them, since it promises the generation and 
making of what has not before existed, e.g. the construction of an 
equilateral triangle, or the describing of a square on a given straight 
line, or the placing of a straight line at a given point. Hence they 
say it is better to assert that all (propositions) are of the same kind, 
and that we r^ard the generation that takes place in them as 
referring not to actual making but to knowledge, when we treat things 
existing eternally as if they were subject to becoming: in other words, 
we may say that everything is treated by way of theorem and not 
by way of problem* l^aina Oeo^pfffiarncw aXX* ov wpofiXfffiarucih 

•* Others on the contrary, like the mathematicians of the school 
of Menaechmus, thought it right to call them all problems, describing 
their purpose as twofold, namely in some cases to furnish {iropi- 
aaaOai) the thing sought, in others to take a determinate object 
and see either what it is, or of what nature, or what is its property, 
or in what relations it stands to something else. 

''In reality both assertions are correct Speusippus is right 
because the problems of geometry are not like those of mechanics, 
the latter being matters of sense and exhibiting becoming and change 
of every sort. The school of Menaechmus are right also because the 
discoveries even of theorems do not arise without an issuing-forth 
into matter, by which I mean intelligible matter. Thus forms going 
out into matter and giving it shape may fairly be said to be like 
processes of becoming. For we say that the motion of our thought 
and the throwing-out of the forms in it is what produces the figures 
in the imagination and the conditions subsisting in them. It is in 
the imagination that constructions, divisions, placings, applications, 
additions and subtractions (take place), but everything in the mind is 
fixed and immune from becoming and from every sort of change'." 
/ " Now those who distinguish the theorem from the problem say 
that every problem implies the possibility, not only of that which is 
predicated of its subject-matter, but also of its opposite, whereas 
every theorem implies the possibility of the thing predicated but not 
of its opposite as well. By the subject-matter I mean the genus 
which is the subject of inquiry, for example, a triangle or a square 
or a circle, and by the property predicated the essential attribute, 
as equality, section, position, and the like, v When then any one 

* Prodas, p. 77, 7— n. • ML pp. 77, 15—78, 8. 

» Md. pp. 78, 8—79, a. 



tt6 INTRODUCTION - t^^«;|4 

enunciates thus, To inscrtbt an equilateral triangle in a circle, he states 
a problem \ for it is also possible to inscribe in it a triangle which 
is not equilateral Again, if we take the enunciation On a given 
limited straight line to construct an equilateral triangle^ this is a 
problem ; for it is possible also to construct one which is not equi- 
lateral. But, when any one enunciates that In isosceles triangles the 
angles at the base are equal, we must say that he enunciates a theorem ; 
for it is not also possible that the angles at the base of isosceles 
triangles should be unequal. It follows that, if any one were to use 
the form of a problem and say In a semkircle to describe a right angie^ 
he would be set down as no geometer For every angle in a semi* 
circle is rights" 

" Zenodotus, who belonged to the succession of Oenopides, but 
was a disciple of Andron^ distinguished the theorem from the problem 
by the fact that the theorem inquires what is the property predicated 
of the subject-matter in it, but the problem what is the cause of what 
effect {rivo^ ivro^ ri ianv}. Hence too Fosidonius defined the one 
(the problem) as a proposition in which it is inquired whether a thing 
exists or not (ct ctniv ^ fuj), the other (the theorem') as a proposition 
in which it is inquired what (a thing) is or of what nature (ri iarw ^ 
woUv Ti) ; and he said that the theoretic proposition must be put in a 
declaratory form, eg., Any iHangk has two sules (together) greater than 
the remaining side and /n any isosceles triangle the angles at the base 
are equals but that we should state the problematic proposition as if 
inquiring whether it is possible to construct an equilateral triangle 
upon such and such a straight line. For there is a difference between 
inquiring absolutely and indeterminately {a'nXm tc xal dopiarm^) 
whether there exists a straight line from such and such a point at 
right angles to such and such a straight line and investigating which 
is the straight line at right angles V 

** That there is a certain difference between the problem and the 
theorem is clear from what has been said ; and that the Elements of 
Euclid contain partly problems and partly theorems will be made 
manifest by the individual propositions, where Euclid himself adds at 
the end of what is proved in them, in some cases^ 'that which it was 
required to do/ and in others, ' that which it was required to prove/ 
the latter expression being regarded as characteristic of theorems, in 
spite of the fact that, as we have said, demonstration is found in 
problems also. In problems, however, even the demonstration is for 
the purpose of (confirming) the construction : for we bring in the 
demonstration in order to show that what was enjoined has been 
done ; whereas in theorems the demonstration is worthy of study for 
its own sake as being capable of putting before us the nature of the 
thing sought. And you will find that Euclid sometimes interweaves 
theorems with problems and employs them in turn, as in the first i 

^ Prodns, pp.79, II— 80,5- f 

* In the text we have t6 di wftfthiiuL answering torhith withoot snbstantive : wphfikmuk . 
was obviously inserted in enor. 

• Froclua, pp. 80. 15—81, 4. 



CH. IX. §4] THEOREMS AND PROBLEMS 127 

book, while at other times he makes one or other preponderate. 
For the fourth book consists wholly of problems, and the fifth of 
theorems V* 

Again, in his note on Eucl. I. 4, Proclus says that Carpus, the 
writer on mechanics, raised the question of theorems and problems in 
his treatise on astronomy. Carpus, we are told, *' says that the class 
of problems is in order prior to theorems. For the subjects, the 
properties of which are sought, are discovered by means of problems. 
Moreover in a problem the enunciation is simple and requires no 
skilled intelligence; it orders you plainly to do such and such a 
thing, to construct an equilateral triangle, or, given two straight lines, to 
cut off from the greater (a straight line) equal to the lesser, and what is 
there obscure or elaborate in these things ? But the enunciation of a 
theorem Ls a matter of labour and requires much exactness and 
scientific judgment in order that it may not turn out to exceed or 
fall short of the truth ; an example is found even in this proposition 
(I. 4), the first of the theorems. Again, in the case of problems, one 
general way has been discovered, that of analysis, by following which 
we can always hope to succeed ; it is this method by which the more 
obscure problems are investigated. But, in the case of theorems, the 
method of setting about them is hard to get hold of since ' up to our 
time,' says Carpus, * no one has been able to hand down a general 
method for their discovery. Hence, by reason of their easiness, the 
class of problems would naturally be more simple.' After these 
distinctions, he proceeds: 'Hence it is that in the Elements too 
problems precede theorems, and the Elements begin from them ; the 
first theorem is fourth in order, not because the fifth* is proved from 
the problems, but because, even if it needs for its demonstration none 
of the propositions which precede it, it was necessary that they should 
be first because they are problems, while it is a theorem. In fact, in 
this theorem he uses the common notions exclusively, and in some 
sort takes the same triangle placed in different positions; the 
coincidence and the equality proved thereby depend entirely upon 
sensible and distinct apprehension. Nevertheless, though the demon- 
stration of the first theorem is of this character, the problems properly 
preceded it, because in general problems are allotted the order of 
precedence'.'" 

Proclus himself explains the position of Prop. 4 after Props, i — 3 
as due to the fact that a theorem about the essential properties of 
triangles ought not to be introduced before we know that such a 
thing as a triangle can be constructed, nor a theorem about the 
equality of sides or straight lines until we have shown, by constructing 
them, that there can be two straight lines which are equal to one 
another^ It is plausible enough to argue in this way that Props. 2 
and 3 at all events should precede Prop. 4. And Prop, i is used in 

* Proclus, p. 81, 5 — M. 

* rh rifiwTow, This should apparently be the fourth because in the next words it is 
implied that none of the first three propositions are required in prorii^ it. 

* Proclus, pp. 141, 19 — 143, II. * ibid, pp. 333, ai— 134, 0. 



138 INTRODUCTION [CB. n. f 4 

Prop. 2, and must therefore precede it But Prop, i showing how to 
construct an equilateral triangle on a given base is not important, in 
relation to Prop. 4, as dealing mth the ** production of triangles " in 
general : for it is of no use to say, as Proclus does, that the construc- 
tion of the equilateral triangle is "common to the three species (of 
triangles)\'' as we are not in a position to know this at such an early 
stage. The existence of triangles in general was doubtless assumed as 
following from the existence of straight lines and points in one plane 
and from the possibility of drawing a straight line from one point to 
another. 

Proclus does not however seem to reject definitely the view of 
Carpus, for he goes on* : ** And perhaps problems are in order before 
theorenis, and especially for those who need to ascend from the arts 
which are concerned with things of sense to theoretical investigation. 
But in dignity theorems are prior to problems.... It is then focuish to 
blame Geminus for saying that the theorem is more perfect than the 
problem. For Carpus himself gave the priority to problems in respect 
of order^ and Geminus to theorems in point of more perfect eSgrnty^ 
so that there was no real inconsistency between the twa 

Problems were classified according to the number of their possible 
solutions. Amphinomus said that those which had a unique solution 
(uorax»?) were called ''ordered'' (the word has dropped out in 
Proclus, but it must be rvt9tiyA»a^ in contrast to the tfiird kind» 
iraicr€L)\ those which had a ddfinite number of solutions "inter- 
mediate " (jkkfra) ; and those with an infinite variety of solutions " un- 
ordered" (£Taicra)^ Proclus gives as an example of the last tihe 
problem To divide a given straight line into three farts in continued 
proportion^. This is the same thing as solving the equations X'\'y-\'Z^a^ 
xz «>*. Proclus' remarks upon the problem show that it was solved, 
like all quadratic equations, by the method of '* application of areas." 
The straight line a was first divided into any two parts, (x-Vz) and j^, 
subject to the sole limitation that {x-Vz) must not be less than 2y^ 
which limitation is the SiopurfjLo^, or condition of possibility. Then 
an area was applied to (;r+ir), or'(a~'y\ ^'falling short by a square 
figure" (fiCKelirov elSei rerpaywv^) and equal to the square on y. This 
determines x and z separately m terms of a and y. For, if ir be the 
side of the square by whiph the area (i.e. rectangle) " falls short," we 
have {(a --y) -z]z ->•, whence 2z « (a -y) ± •J [{a -yY - 4j/^}. And 
y may be chosen arbitrarily, provided that it is not greater than a/3. 
Hence there are an infinite number of solutions. U y^aji, then, as 
Proclus remarks, the three parts are equal. 

Other distinctions between different kinds of problems are added 
by Proclus. The word " problem," he says, is used in several senses. 
In its widest sense it may mean anything " propounded " (^/Kn-ciyd- 
fuvov\ whether for the purpose of instruction (jLoBrjatm) or construc- 
tion (^oii}atfa>9). (In this sense, therefore, it would include a theorem.) 

^ Prodns, p. 934, 91. * ibU. p. 143, 19—95. 

* ihid, p. 990^ 7 — 19. ^ ibid. pp. 990^ 16—991, 6. 



I 



CH. ix.§4] THEOREMS AND PROBLEMS 139 

But its special sense in mathematics is that of something *' propounded 
with a view to a theoretic construction'." 

Again you may apply the term (in this restricted sense) even to 
something which is impossible, although it is more appropriately used 
of what is possible and neither asks too much nor contains too little in 
the shape of data. According as a problem has one or other of these 
defects respectively, it is called (i) a problem in excess (irXeovd^op) or 
(2) a deficient problem (jKKvirh irpopkruia). The problem in excess 
(i) is of two kinds, {a) a problem in which the properties of the 
figure to be found are either inconsistent {aavfifiara) or non-existent 
(avvTrap/era), in which case the problem is called impossible, or (b) a 
problem in which the enunciation is merely redundant : an example 
of this would be a problem requiring us to construct an equilateral 
triangle with its vertical angle equal to two-thirds of a right angle ; 
such a problem is possible and is called "more than a problem" (fiei^op 
^ irpofikfffia). The deficient problem (2) is similarly called " less than 
a problem " (IXaar.aop fj wp6fi\fffia), its characteristic being that 
something has to be added to the enunciation in order to convert it 
from indeterminateness (aopurria) to order (rof w) and scientific deter- 
minateness (Spo^ hriarrffiopuco^) : such would be a problem bidding 
you " to construct an isosceles triangle," for the varieties of isosceles 
triangles are unlimited. Such ''problems" are not problems ki the 
proper sense {Kvplw^ Xeyofiepa wpol3Xf}fiaTa), but only equivocally". 

§ 5. THE FORMAL DIVISIONS OF A PROPOSITION. 

"Every problem," says Proclus', "and every theorem which is 
complete with all its parts perfect purports to contain in itself all of 
the following elements: enunciation (wporaai^), setting-out (licOeai^), 
definition or specification {SiopuTfi6^\ construction or machinery 
(tearaa-icevi^), proof (dwo&eifi^), conclusion (avfiwipao'fia). Now of 
these the enunciation states what is given and what is that which is 
sought, the perfect enunciation consisting of both these parts. The 
setting-out marks off what is given, by itself, and adapts it before- 
hand for use in the investigation. The definition or specification 
states separately and makes clear what the particular thing is which 
is sought. The construction ox machinery adds what is wanting to the 
datum for the purpose of finding what is sought. The pro^ draws 
the required inference by reasoning scientifically from acknowledged 
facts. The conclusion reverts again to the enunciation, confirming 
what has been demonstrated. These are all the parts of problems 
and theorems, but the most essential and those which are found in all 
, are enunciation, proof, conclusion. For it is equally necessary to know 
beforehand what is sought, to prove this by means of the intermediate 
steps, and to state the proved fact as a conclusion ; it is impossible 
to dispense with any of these three things. The remaining parts 
are often brought in, but are often left out as serving no purpose. 

^ Proclus, p. an, 7 — 11. ■ ihid, pp. an, 13 — aaa, 14. 

• ibid. pp. ao3, 1—104, 13 ; 104, a3— «05» 8. 

O 



H. E. 



130 INTRODUCTION [ch. «• 1 5 

Thus there is neither settiftg-ami nor defittUum in the problem of 
constructing an isosceles triangle having each of the angles at the 
base double of the remaining angle, and in most theorems there 
is no construction because the setting-out suffices without any addition 
for proving the required property from the data. When then do 
we say that the setting-cut is wanting ? The answer is, when there 
is nothing given in the enunciation \ for, though the enunciation is 
in general divided into what is given and what is sought, this 
is not always the case, but sometimes it states only what b sought, 
i.e. what must be known or found, as in the case of the problem 
just mentioned. That problem does not, in (act, state befordumd | 
with what datum we are to construct the isosceles triangle having j 
each of the equal angles double of the remaining angle, but (simply) | 
that we are to find such a triangle.... When, then, the enuncia- 
tion contains both (what is given and what is sought), in that case 
we find both definition and setting-out^ but, whenever the datum 
is wanting, they too are wanting. For not only is the settit^-out 
concerned with the datum, but so is the definition also^ as, in the 
absence of the datum, the definition will be identical with the 
enunciation. In fact, what could you say in defining the object of 
the aforesaid problem except that it is required to find an isosceles 
triangle of the kind referred to? But that is what the enundaiioH 
stated. If then the enunciation does not include, on the one hand, 
what is given and, on the other, what is sought, there is no setting-out 
in virtue of there being no datum, and the definition is left out in 
order to avoid a mere repetition of the enumiation!^ 

The constituent parts of an Euclidean proposition will be readily 
identified by means of the above description. As regards the defi- 
nition or specification (Biopicfio^) it is to be observed that we have 
here only one of its uses. Here it means a closer definition or descrip- 
tion of the object aimed at, by means of the concrete lines or figures 
set out in the exOeai^ instead of the general terms used in the enun- 
ciation ; and its purpose is to rivet the attention better, as Proclus 
indicates in a later passage {rpoirov rwh irpoaex^la^ iarlv alrio^ o 
htopiaiUs^y. 

The other technical use of the word to signify the limitations to 
which the possible solutions of a problem are subject is also described 
by Proclus, who speaks of Biopia^ioi determining ''whether what is 
sought is impossible or possible, and how far it is practicable and in 
how many ways'" ; and the Biopurfio^ in this sense appears in Euclid 
as well as in Archimedes and Apollonius. Thus we have in Eucl. I. 
22 the enunciation *'From three straight lines which are equal to 
three given straight lines to construct a triangle," followed imme- « 
diately by the limiting condition (Bioptafi6<;). "Thus two of the 
straight lines taken together in any manner must be greater than the 
remaining one." Similarly in VI. 28 the enunciation "To a given 
straight line to apply a parallelogram equal to a given rectilineal 

> Proclus, p. 108, II. * Mii. p. 303, 3. 



CH. IX. is] FORMAL DIVISIONS OF A PROPOSITION 131 

figure and falling short by a parallelogrrammic figure similar to a 
given one " is at once followed by the necessary condition of possi- 
bility: "Thus the given rectilineal figure must not be greater than 
that described on half the line and similar to the defect." 

Tannery supposed that, in giving the other description of the 
hiopwiU^ as quoted above, Proclus, or rather his g^ide, was using the 
term incorrectly. The SiopiafAo^ in tlie better known sense of the 
determination of limits or conditions of possibility was, we are told, 
invented by Leon. Pappus uses the word in this sense only. The 
other use of the term might. Tannery thought, be due to a confusion 
occasioned by the use of the same words {Set £17) in introducing the 
parts of a proposition corresponding to the two meanings of the word 
Siopur/AOf;^ On the other hand it is to be observed that Eutocius 
distinguishes clearly between the two uses and implies that the differ- 
ence was well known*. The Biopiafiif: in the sense of condition of 
possibility follows immediately on the enunciation, is even part of it ; 
the Biopiafio^ in the other sense of course comes immediately after the 

Proclus has a useful observation respecting the concltision of a 
proposition'. "The conclusion they are accustomed to make double 
in a certain way : I mean, by proving it in the given case and then 
drawing a general inference, passing, that is, from the partial con- 
clusion to the general. For, inasmuch as they do not make use of 
the individuality of the subjects taken, but only draw an angle or a 
straight line with a' view to placing the datum before our eyes, they 
consider that this same fact which is established in the case of the 
particular figure constitutes a conclusion true of every other figure of 
the same kind. They pass accordingly to the general in order that 
we may not conceive the conclusion to be partial. And they are 
justified in so passing, since they use for the demonstration the par- 
ticular things set out, not qud particulars, but qud typical of the rest 
For it is not in virtue of such and such a size attaching to the angle 
which is set out that I effect the bisection of it, but in virtue of its 
being rectilineal and nothing more. Such and such size is peculiar to 
the angle set out, but its quality of being rectilineal is common to all 
rectilineal angles. Suppose, for example, that the given angle is a 
right angle. . If then I had employed in the proof the fact of its being 
right, I should not have been able to pass to every species of recti- 
lineal angle ; but, if I make no use of its being right, and only consider 
it as rectilineal, the argument will equally apply to rectilineal angles 
in general." 

' La Ciomitrie grecque^ p. 149 note. Where det 9^ introduces the closer description of 
the problem we may translate, "it is then required** or '*thus it is required" (to construct etc): 
when it introduces the condition of possibility we may translate **thus it is necessary etc.*' 
Heiberg originally wrote dct M in the latter sense in I. 13 on the authority of Produs and 
Eutocius, and against that of the Mss. Later, on the occasion of xi. 33, he observed that he 
should have followed the mss. and written dct ^ which he found to be, after all, the right 
reading in Eutocius (Apollonius, ed. Heiberg, 11. p. 178). det ^ is also the expression usied 
by Diophantus for introducing conditions of possibility. 

' See the passage of Eutocius referred to in last note. ' Proclus, p. 107, 4 — 15. 

9—2 



132 INTRODUCTION [cb. dl |6 

§ 6. OTHER TECHNICAL TERMS. 

I. Things said to be given. 

Proclus attaches to his description of the formal ^ivisiona of a 
proposition an explanation of the different senses in which the word 
given or datum (BeSofiit^ov) is used in geometry. ''Everything that is 
given is given in one or other of the following ways^ in fosiiiam^ in 
ratio, in magnitude, or in specks. The point is given in pasiium only, 
but a line and the rest may be given in all the senses^" 

The illustrations which Proclus gives of the four senses in which a 
thing may be given are not altogether happy, and, as regards things 
which are given in position^ in fn^gnitude^ and in species, it is best, I 
think, to follow the definitions given by Euclid himself in his book of 
Data. Euclid does not mention the fourth class, things given in ratio^ 
nor apparently do any of the great eeometers. 

( 1 ) Given in position really needs no definition ; and, when Euclid 
says (Data, Def 4) that ^Points, lines and angles are said to ht given 
in position which always occupy the same place,| we are not really 
the wiser. 

(2) [Given in magnitude is defined thus {Data^ De£ l): ''Aieas, 
lines and angles are called given in tnagnitude to which we can find 
equals."! Proclus' illustration is in this case the following: when, he 
says, two unequal straight lines are given from the greater of which 
we have to cut ofT a straight line equal to the lesser, & straight lines 
are obviously given in magnitude^ *' for greater and less, and finite 
and infinite are predications peculiar to magnitude." But he does not 
explain that part of the implication of the term is that a thing is given 
in magnitude only, and that, for example, its position is not given and 
is ajnatter of indifference. 

j(3) Given in species. Euclid's definition {Data, Def. 3) is: 
'' Rectilineal figures are said to b^ given in species in which the angles 
are severally given and the ratios of the sides to one another are 

given/' I And this is the recognised use of the term (cf. Pappus, 

passim). Proclus uses the term in a much wider sense for which I am 
not aware of any authority. Thus, he says, when we speak of (bisect- 
ing) a given rectilineal angle, the angle is given in species by the word 
rectilineal, which prevents our attempting, by the same method, to 
bisect a curvilineal angle ! On Eucl. i. 9, to which he here refers, he 
says that an angle is given in species when e.g. we say that it is right 
or acute or obtuse or rectilineal or " mixed," but that the actual angle 
in the proposition is given in species only. As a matter of fact, we 
should say that the actual angle in the figure of the proposition is 
given in magnitude and not in species, part of the implication of given 
in species being that the actual magnitude of the thing given in species 
is indifferent ; an angle cannot be given in species in this sense at a^. 
The confusion in Proclus' mind is shown when, after saying that a 
right angle is given in species, he describes a third of a right angle as 
given in magnitude. 

' Prodiu, p. «05, 13-^15. 



CH. ix.§6] OTHER TECHNICAL TERMS 133 

No better example of what is meant by given in species^ in its 
proper sense, as limited to rectilineal figures, can be quoted than the 
g^ven parallelogram in Eucl. VI. 28, to which the required parallelo- 
gram has to he made similar; the former parallelogram is in fact 
given in species^ though its actual size, or scale, is indifferent 

(4) I Given in ratio presumably means something which is given 
by means of its ratio to some other given thing.( This we gather from 
Proclus' remark (in his note on I. 9) that an angle may be given in 
ratio ** as when we say that it is double and treble of such and such an 
angle or, generally, greater and less." The term, however, appears to 
have no authority and to serve no purpose. Proclus may have 
derived it from such expressions as "in a given ratio" which are 
common enough. 

2. Lemma. 

**The term Umma" says Proclus*, "is often used of any proposition 
which is assumed for the construction of something else : thus it is a 
common remark that a proof has been made out of such and such 
lemmas. But the special meaning of lemma in geometry is a 
proposition requiring confirmation. For when, in either construction 
or demonstration, we assume anything which has not been proved but 
requires argument, then, because we regard what has been assumed as 
doubtful in itself and therefore worthy of investigation, we call it a 
lemma\ differing as it does from the postulate and the axiom in being 
matter of demonstration, whereas they are immediately taken for 
granted, without demonstration, for the purpose of confirming other 
things. Now in the discovery of lemmas the best aid is a mental 
aptitude for it. For we may see many who are quick at solutions and 
yet do not work by method ; thus Cratistus in our time was able to 
obtain the required result from first principles, and those the fewest 
possible, but it was his natural gift which helped him to the discovery. 

* Proclus, pp. ail, i — an, 4. 

* It would appear, sajrs Tannery (p. 15111.), that Geminus understood a lemma as being 
simply XcMi/Sar^eror, something assumed (cf. the passage of Proclus, p. 73, 4, relating to 
Menaechmus' view of eiements) : hence we cannot consider ourselves authorised in attributing 
to Geminus the more technical definition of the term here given by Proclus, according to 
which it b only used of propositions not proved beforehand. This view of a lemma must 
be considered as relatively modem. It seems to have had its origin in an imperfection of 
method. In the course of a demonstration it was necessary to assume a proposition which 
required proof, but the proof of which would, if inserted in the particular place, break the 
thread ot the demonstration : hence it was necessary either to prove it beforehand as a 
preliminary proposition or to postpone it to be proved afterwards {in i^ 5ctx^cra«). 
When, after the time of Geminus, the progress of oneinal discovery in geometry was arrested, 
geometers occupied themselves with the study and elucidation of the works of the great 
mathematicians who had preceded them. This involved the investigation of propositions 
explicitly quoted or tsicitly assumed in the great classical treatises; and naturally it was found 
that several such remained to be demonstrated, either because the authors had omitted 
them as being easy enough to be left to the reader himself to prove, or because books in 
irhich they were proved had been lost in the meantime. Hence arose a class of complementary 
or auxiliary propositions which were called iemptas. Thus Pappus gives in his Book vn a 
collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the 
so-called "Treasury of Analysis " (r6rot dvaXu^erot). When Proclus goes on to distinguish 
three methods of discovering lemmas, analysis^ dnnsicn, and reductic ad adsurditmf he seems 
to imply that the principal business of contemporary geometers was the investigation of these 
auxiliary propositions. 



Iu~ 



134 INTRODUCTION [CB. n. |6 

Nevertheless certain methods have been handed down. The finest b 
the method which by means of attafysis carries the thing aought up to 
an acknowledged principle, a method which Plato, as tney say. ccMn- 
municated to Leodamas\ and by which the latter, too, is flAid to have 
discovered many things in geometry. The second is tiie method of 
division^ which divides into its parts the genus proposed for con- 
sideration and gives a starting-point for the demonstration b^ means 
of the elimination of the other elements in the construction of what is 
proposed, which method also Plato extolled as being of assistance to 
all sciences. The third is that by means of the ndMctw ad aisunbim^ 
which does not show what is sought directly, but refutes its opposite 
and discovers the truth incidentaUy.** 

3. Case. 

"^ The case^ (wrScif:)" Proclus proceeds^ ''announces difierent ways 
of construction and alteration <^ positions due to the transposition of 
points or lines or planes or solidk And, in general, all its varieties 
are seen in the figure, and this is why it b called our, being a trans- 
position in the construction." 

4. Porism. 

** The term porism is used also <^ certain problems such as the 
Porisms written by Euclid. But it is specially used when from what 
has been demonstrated some otiher theorem b revealed at the same 
time without our propounding it which theorem has on thb very 
account been called a porism (corollary) as being a sort of inddenttd 
gain arising from the scientific demonstration*." Cf the note on L 15. 

* This punge and another from Dioeenes Laertias (ill. 34, p. 7^ ed. Cobet) to the cflect 
that *' He [Plato] explained (tlnrrh^^r^ to Lcodamas of Thasos the method of inquiry by 
analysis " have been commonly understood as ascribing to Plato the invention of the method 
of analvsts ; but Tannery points out forcibly (pp. 1 ta, 113) how difficult it is to explain in 
what Plato's discovery could have consisted if tmalfsis be taken in the sense attributed to it 
In Pappus, where we can see no more than a series of successive reductions of a problem 
until It is finally reduced to a known problem. On the other hand, Proclus' words about • 
carrying up the thing sought to *' an acknowledged principle " suggest that what he had in 



mind was the process described at the end of Book vi of the Republic by which the dialec< 
tidan (unlike tne mathematician) uses hypotheses as stepping-stones np to a principle which 
is not hypothetical, and then is able to descend step Dy step verifying every one of the 
hypotheses by which he ascended. This description does not of course rder to mathematical 
analvsis, but it may have given rise to the ioea that analysis was Plato's discovery, since 
anafytis and synthesis following each other are related in the same way as the upward and 
the downward progression in the dialectician's intdlectual method. And it may be that 
Plato's achievement was to observe the importance, from the pomt of view of logiod rigour, 
of the confirmatory synthesis following analysis, and to regularise in this way and elevate 
Into a completely irrefragable method the partial and uncertain analysis upon which the 
works of his predecessors depended. 

* Here apin the successive bipartitions of genera into species such as we find in the 
Sophist and Republic have very little to say to geometry, and the very fact that they are hoe 
mentioned side bv side with analysis suggests that Proclus confused the latter with the 
philosophical method of Ref. vi. 

* Tannery rightly remarks (p. 15a) that the subdivision of a theorem or problem Into 
several cases is foreign to the really classic form ; the ancients preferred, where necessary, to 
multiply enunciations. As, however, some omissions necessarily occurred, the writers of 
lemmas naturally added separate casa, which in some instances found their way into die text. 
A good example is Eudid i. 7, the second case of which, as it appears in our text-books, 
was interpolated. On the commentary of Proclus on this proposition Th. Taylor rightly 
remarks that *' Euclid everywhere avokfs a multitude of cases." 

* Produs, p. SIS, 5— II. 

* Tannery notes however that, so fin'^lrom distingutshing his corollaries from the con- 



CH. ix.§6] OTHER TECHNICAL TERMS 135 

5. Objection. 

" The objectuni (Ivaraa-i^) obstructs the whole course of the argu- 
ment by appearing as an obstacle (or crying ' halt/ ehravTwaa) either 
to the construction or to the demonstration. There is this difference 
between the objection and the case, that, whereas he who propounds 
the case has to prove the proposition to be true of it, he who makes 
the objection does not need to prove anything : on the contrary it is 
necessary to destroy the objection and to show that its author is 
saying what is false*." 

That is, in general the objection endeavours to make it appear that 
the demonstration is not true in every case ; and it is then necessary 
to prove, in refutation of the objection, either that the supposed case 
is impossible, or that the demonstration is true even for that case. A 
good instance is afforded by Eucl. I. 7. The text- books give a second 
case which is not in the original text of Euclid. Proclus remarks on 
the proposition as given by Euclid that the objection may conceivably 
be raised that what Euclid declares to be impossible may after all be 
possible in the event of one pair of straight lines falling completely 
within the other pair. Proclus then refutes the objection by proving 
the impossibility in that case also. His proof then came to ht given 
in the text-books as part of Euclid's proposition. 

The objection is one of the technical terms in Aristotle's logic and 
its nature is explained in the Prior Analytics^ "An objectiofi is a 
proposition contrary to a proposition.... Objections are of two sorts, 
general or partial.... For when it is maintained that an attribute 
belongs to every (member of a class), we object either that it belongs 
to none (of the class) or that there is some one (member of the class) 
to which it does not belong." 

6. Reduction. 

This is again an Aristotelian term, explained in the Prior 
Analytics^ It is well described by Proclus in the following passage : 

" Reduction {dwayaryi]) is a transition from one problem or theorem 
to another, the solution or proof of which makes that which is pro- 
pounded manifest also. For example, after the doubling of the cube 
had been investigated, they transformed the investigation into another 
upon which it follows, namely the finding of the two means ; and from 
that time forward they inquired how between two given straight lines 
two mean proportionals could be discovered. And they say that the 
first to effect the reduction of difficult constructions was Hippocrates of 
Chios, who also squared a lune and discovered many other things in 
geometry, being second to none in ingenuity as r^ards constructions*." 

dnsioDS of his propositions, Euclid inserts them before the closing words "(being) what it 
was required to do*' or "to prove.** In fact the porism-corollary is with Euclid rather a 
modifira form of the regular conclusion than a separate proposition. 



* Proclus, p. 211, 18 — 13. 
■ ^na/. /nor. ii. 26, 69 a 37. 



' idid, II. 35, 69 a 10. 



* Proclus, pp. 113, 14—313, II. This passage has frequently been taken as crediting 
Hippocrates with the discovery of the method of geometrical reduction : cL Taylor (Transla- 
tion of Proclus, II. p. 16), Allman (p. 41 n., 59), Gow (pp. 169, 170). As Tannery remarks 
(p. 110), if the particular reduction of the duplication problem to that of the two means is 



13d INTROftUCtlOK {^ hu ii 

7, Reductio ad absurdum. 

This is variously called by Aristotle "rediuiia ad absurdum^' (^ m 
nJ m^aTov awaywy ri)\ *" proof /^r impossibik'^ {i} Zia rov aSvudrau 
Bei^i^i or airoSfif*?)*, "proof leading to the impossible" {^ ca9 to 
tiBvparoif ayov^a fhraBeih^y. It is part of *' proof (starting) from a 
hypothesis**' (cf vwo0€iJ€oi^). "All (syllogisms) which reach the 
conclusion fier imfossibili reason out a conclusion which is false, and 
they prove the original contention (by the method starting) from a 
hypothesis, when something impossible results from assuming the 
contradictory of the original contention, as, for example, when it is 
proved that the diagonal (of a square) is incommensurable because, 
if it be assumed commensurable, it will follow that odd (numbers) 
are equal to even (numbers)*/* Or again, *' proof (leading) to the 
impossible differs from the direct (S€*<t*«^5) in that it assumes what 
it desires to destroy [namely the hypothesis of the falsity of the 
conclusion] and then reduces it to something admittedly false, whereas 
the direct proof starts from premisses admittedly true'.'* 

Proclus has the following description of the reductw ad absurdum. 
'* Proofs by reductio ad ahsurdum \n every case reach a conclusion 
manifestly impossible, a conclusion the contradictory of which is 
admitted. In some cases the conclusions are found to conflict with 
the common notions, or the postulates, or the hypotheses (from which 
we started) ; in others they contradict propositions previously estab- 
lished^".*/' Every reducth ad absurdum assumes what conflicts with 
the desired result, then, using that as a basis, proceeds until it arrives 
at an admitted absurdity, and^ by thus destroying the hypothesis, 
establishes the result originally desired. For it is necessary to under- 
stand generally that all mathematical arguments either proceed from 
the first principles or lead back to them, as Porphyry somewhere says. 
And those which proceed from the hrsX principles are again of two 
kinds, for they start cither from common notions and the clearness of 
the self-evident alond or from results previously proved \ while those 
which lead back to the principles are either by way of assuming the 
principles or by way of destroying them. Those which assume the 
principles are called analyses^ and the opposite of these are sjnfAeses — 
for it is possible to start from the said principles and to proceed in 
the regular order to the desired conclusion, and this process is syn- 
ihesis — while the arguments which would destroy the principles are 

the first noted in hisioryj it « difficult to £tipf>o*ic that it was really the firet ; for Hippocrates 
x&u&l have found instances of ii in the Pythi^ortjji geomctiy- lirelschneidei, I ihiiiL, tonics 
nearer the truth when he boldly (p. 99) translates: "Tms reduction c/tie t/artsaiii ram- 
struct ion is said to have been fint given by Hippocrates.'* The words are wfitrntf Ik fun 
xAr Aropavfidwm BtaypofiMrif rV lva7«ryV voci^o^^t which must, literally, be translated 
as in the text above; but, when Proclus speaks vaguely of "difficult oonstractioiis,'' he 
probably means to say simply that '* this firrt recorded instance of a reduction of a difficult 
construction is attributed to Hippocrates."' 

I Aristotle, Anai.frwr, i. 7t ^9 b 5 ; i. 44, 50 a 30. 

* Uid. I. 91, 39 b 51 ; I. ap, 45 a 55. 

* Anai, past. I. 94, 85 a 10 etc. ^ Anal, prior, i. 93, 40 b 95. 

* Anal. prUr. I. 93, 41 a 94. * iHd. 11. 14, 69 b 99. 
^ Produs, p. 954, 99—97. 






r 



CH. IX.46] OTHER TECHNICAL TERMS 137 

called reductiones ad absurdum. For it is the function of this method 
to upset something admitted as clear*." 

8. Analysis and Synthesis. 

It will be seen from the note on Eucl. XIII. i that the MSS. of the 
Elements contain definitions of Analysis and Synthesis followed by 
alternative proofs of xill. i — 5 after that method. The definitions and 
alternative proofs are interpolated, but they have great historical 
interest because of the possibility that they represent an ancient 
method of dealing with these propositions, anterior to Euclid. The 
propositions give properties of a line cut "in extreme and mean ratio," 
and they are preliminary to the construction and comparison of the 
five regular solids. Now Pappus, in the section of his CoUection dealing 
with the latter subject*, says that he will give the comparisons between 
the five figures, the pyramid, cube, octahedron, dodecahedron and 
icosahedron, which have equal surfaces, " not by means of the so-called 
analytical vdK^VTf, by which some of the ancients worked out the proofs, 
but by the synthetical method*...." The conjecture of Bretschneider 
that the matter interpolated in Eucl. xill. is a survival of investiga- 
tions due to Eudoxus has at first sight much to commend it^ In the 
first place, we are told by Proclus that Eudoxus " greatly added to 
the number of the theorems which Plato originated regarding the 
section^ and employed in them the method of analysis'.** It is obvious 
that " the section " was some particular section which by the time of 
Plato had assumed great importance ; and the one section of which 
this can safely be said is that which was called the " golden section," 
namely, the division of a straight line in extreme and mean ratio 
which appears in Eucl. II. 1 1 and is therefore most probably Pytha- 
gorean. Secondly, as Cantor points out*, Eudoxus was the founder 
of the theory of proportions in the form in which we find it in Euclid 
v., VI., and it was no doubt through meeting, in the course of his 
investigations, with proportions not expressible by whole numbers 
that he came to realise the necessity for a new theory of proportions 
which should be applicable to incommensurable as well as commen- 
surable magnitudes. The "golden section" would furnish such a case. 
And it is even mentioned by Proclus in this connexion. He is 
explaining' that it is only in arithmetic that all quantities bear 
"rational" ratios (/J»;toc >jlrio^) to one another, while in geometry there 
are '* irrational " ones (apptfro^) as well. " Theorems about sections 
like those in Euclid's second Book are common to both [arithmetic 
and geometry] except that in which the straight line is cut in extreme 
and mean ratio^** 

» Proclus, p. 355, 8— «6. 

■ Pappus, V. p. 410 sqq. • ibid. pp. 410, 17 — 4H, a. 

^ Bretschneider, p. 108. See however Heibeig's recent suggestion (Paralipomena tu 
EukUd in Hermes^ xxxviii., 1003) that the author was Heron. The suggestion is based 
on a comparison with the remarks on analysis and synthesis Quoted from Heron by an-NairizI 
(ed. Cortxe, p. 89) at the beginning of his commentary on Eucl. Book il. On the whole, 
this suggestion commends itself to me more than that of Bretschneider. 

* Proclus, p. 67, 6. * Cantor, Gtsch, d, Maih, if, p. 941. 

' Proclus, p. 60, 7 — 9. • ibid, p. 60, 16 — 19. 



138 INTRODUCTION [cB.a.f6 

The definitions of Analysis and Symthssis interpolated in EucL 
XIII. are as follows (I adopt the reading of B and V, the only in- 

telligible one, for the second^ 

" Analysis is an assumption of that which is sought as if it were 
admitted < and the passage > through its consequences to sbmething 
admitted (to be) true. 

" Synthesis is an assumption of that which is admitted < and the 
passage > through its consequences to the finishing or attainment of 
what is sought" 

The language is by no means clear and has, at the best, Id be 
filled out. 

Pappus has a fuller account* : 

"* The so-called avakuiiktya^ (' Treasury of Analysis ') is. to put it 
shortly, a special body of doctrine provided for the use of those who. 
after finishing the ordinary Elements, are desirous of acquiring the 
power of solving problems which may be set them involving (the 
construction of) lines, and it is useful for this alone. It is the woric 
of three men, Euclid the author of the Elements, ApoUonius of Perga, 
and Aristaeus the elder, and proceeds by way of analysis and synthesis. 

" Analysis then takes that which is sought as if it were admitted 
and passes from it through its successive consequences to something 
which is admitted as the result of synthesis: for in analysis we assume 
that which is sought as if it were (already) done (tctoi^X ^^'^ ^"^ 
inquire what it is from which this results,^ and again what is the ante- 
cedent cause of the latter, and so on, until by so retracing our steps 
we come upon something already known or belonging to the class of 
first principles, and such a method we call analysis as being solution 
backwards {avdiraKiv \vcivy, 

" But in synthesis, reversing the process, we take as already done 
that which was last arrived at in the analysis and, by arranging in 
their natural order as consequences what were before antecedents, 
and successively connecting them one with another, we arrive finally 
at the construction of what was sought ; and this we call synthesis. 

'' Now analysis is of two kinds, the one directed to searching for 
the truth and called theoretical, the other directed to finding what we 
are told to find and called problematicaL (i) In the theoretical kind 
we assume what is sought as if it were existent and true, after which 
we pass through its successive consequences, as if they too were true 
and established by virtue of our hypothesis, to something admitted : 
then {a\ if that something admitted is true, that which is sought will 
also be true and the proof will correspond in the reverse order to the 
analysis, but {b\ if we come upon something admittedly false, that 
which is sought will also be false. (2) In the problematical kind we 
assume that which is propounded as if it were known, after which we 
pass through its successive consequences, taking them as true, up to 
something admitted : if then {a) what is admitted is possible and 
obtainable, that is, what mathematicians call given^ what was originally 
proposed will also be possible, and the proof will again correspond in 

* PApptts, vii. pp. 634 — 6. 



il 



CH.ix.§6] OTHER TECHNICAL TERMS 139 

reverse order to the ianalysis, but if (d) we come upon something 
admittedly impossible, the problem will also be impossible." 

The ancient Analysis has been made the subject of careful studies 
by several writers during the last half-century, the most complete 
being those of Hankel, Duhamel and Zeuthen ; others by Ofterdinger 
and Cantor should also be mentioned^ 

The method is as follows. It is required, let us say, to prove that 
a certain proposition A is true. We assume as a hypothesis that A 
is true and, starting from this we find that, if A is true, a certain 
other proposition B is true ; if B is true, then C ; and so on until 
we arrive at a proposition K which is admittedly true. The object 
of the method is to enable us to infer, in the reverse order, that, since 
K is true, the proposition A originally assumed is true. Now 
Aristotle had already made it clear that false hypotheses might lead 
to a conclusion which is true. There is therefore a possibility of error 
unless a certain precaution is taken. While, for example, B may be a 
necessary consequence of A, it may happen that A is not a necessary 
consequence of B. Thus, in order that the reverse inference from the 
truth of K that A is true may be logically justified, it is necessary 
that each step in the chain of inferences should be unconditionally 
convertible. As a matter of fact, a very large number of theorems in 
elementary geometry are unconditionally convertible, so that in practice 
the difficulty in securing that the successive steps shall be convertible 
is not so great as might be supposed. But care is always necessary. 
For example, as Hankel says', a proposition may not be uncon- 
ditionally convertible in the form in which it is generally quoted. 
Thus the proposition " The vertices of all triangles having a common 
base and constant vertical angle lie on a circle " cannot be converted 
into the proposition that "All triangles with common base and vertices 
lying on a circle have a constant vertical angle*'; for this is only true 
if the further conditions are satisfied (i) that the circle passes through 
the extremities of the common base and (2) that only that part of the 
circle is taken as the locus of the vertices which lies on one side of the 
base. If these conditions are added, the proposition is unconditionally 
convertible. Or again, as Zeuthen remarks', K may be obtained by 
a series of inferences in which A or some other proposition in the 
series is only apparently used ; this would be the case e.g. when the 
method of modem algebra is being employed and the expressions on 
each side of the sig^ of equality have been inadvenently multiplied 
by some composite magnitude which is in reality equal to zero. 

Although the above extract from Pappus does not make it clear 
that each step in the chain of argument must be convertible in the 
case taken, he almost implies this in the second part of the definition 
of Analysis where, instead of speaking of the consequences B, C... 

^ Hankel, Zur Gtschichte der Mathematik in Altertkum und MUUlalter^ 1 874, pp. 137— 1 50 ; 
Duhamel, Dts mStkodes dans Us sciences de raisonnement^ Part I., 3 ed., Paris, 1885, pp. 39 — 68 ; 
2^then, Gesckickie der Mathematik im Altertum und Mitteiaiter^ 18961 pp. 91 — 104; 
Ofterdinger, Beitrage tmr Gesckickie der grieckiscken Maikimaiik^ Ulm, 1800; Cantor, 
Gesckickie der Matkematik, ij, pp. a 10 — 1. 

' Hankel, p- 139. ' Zeuthen, p. 103. 






I40 INTRODUCTION [CB. OL f 6 

successively following from A» he suddenly changes the expreitkm 
and says that we inquire wkai ii is (B)fivm which A fallows (A bdng 
thus the consequence of B, instead of the reverse), and then what 
(viz. C) is the antecedent cause of B; and in practice tlie Greeks 
secured what was wanted by always insisting on the analysis being 
confirmed by subsequent synthesis, that is, they laboriously worked 
backwards the whole way from K to Ap reversing the order of the 
analysis, which process would undoubtedly bring to light any flaw 
which had crept into the argument through tibe accidental neglect of 
the necessary precautions. 

Reductio ad absurdum a variety of analysis. 

In- the process of analysis starting from the hypothesis that a 
proposition A is true and passing through B, C... as successive con- 
sequences we may arrive at a proposition K which, instead of being 
admittedly true, is either admittedly false or the contradictory of the 
original hypothesis A or of some one or more of the propositions B^ C... 
intermediate between A and K. Now correct inference from a true 
proposition cannot lead to a false ^proposition ; and in this case there- 
fore we may at once conclude, wiuout any inquiry whether the 
various steps in the ailment are convertible or not, that the hypo- 
thesis A is false, for, if it were true, all the consequences correctly 
inferred from it would be true and no incompatibility could arise. 
This method of proving that a given hypothesis is false furnishes an 
indirect method of proving that a given hypodiesis A is true^ since we 
have only to take the contradictory of A and to prove that it is false. 
This is the method of reductio ad absurdum^ which is therefore a variety 
of analysis. The contradictory of A, or not- A, will generally include 
more than one case and, in order to prove its falsity, each of the cases 
must be separately disposed of: e.g., if it is desired to prove that a 
certain part of a figure is equal to some other part, we take separately 
the hypotheses (i) that it is greater^ (2) that it is less^ and prove 
that each of these hypotheses leads to a conclusion either admittedly 
false or contradictory to the hypothesis itself or to some one of its 
consequences. 

Analysis as applied to problems. 

It is in relation to problems that the ancient analysis has the 
greatest significance, because it was the one general method which 
3ie Greeks used for solving all "the more abstruse problems" (rii 
daa^arepa' rAv TrpofiXsffLdrc^py. I 

We have, let us suppose, to construct a figure satisfying a certain 
set of conditions. If we are to proceed at all methodically and not 
by mere guesswork, it is first necessary to "analyse" those conditions. 
To enable this to be done we must get them clearly in our minds, 
which is only possible by assuming all the conditions to be actually 
fulfilled, in other words, by supposing the problem solved. Then we 
have to transform those conditions, by all the means which practice in 
such cases has taught us to employ, into other conditions which are 
necessarily fulfilled if the original conditions are, and to continue this 

^ Prodiu, p. 94*, 16, 17. 



CH. ix.§6] OTHER TECHNICAL TERMS 141 

transformation until we at length arrive at conditions which we 
are in a position to satisfy ^ In other words, we must arrive at 
some relation which enables us to construct a particular part of 
the figure which, it is true, has been hypothetically assumed and 
even drawn, but which nevertheless really requires to h^ found in 
order that the problem may be solved. From that moment the 
particular part of the figure becomes one of the data, and a fresh 
relation has to be found which enables a fresh part of the figure 
to be determined by means of the original data and the new one 
together. When this is done, the second new part of the figure also 
belongs to the data ; and we proceed in this way until all the parts 
of the required figure are found*. The first part of the analysis 
down to the point of discovery of a relation which enables 
us to say that a certain new part of the figure not belonging 
to the original data is given, Hankel calls the transformation ; the 
second part, in which it is proved that all the remaining parts of 
the figure are "given," he calls the resolution. Then follows the 
synthesis^ which also consists of two parts, (i) the construction^ in 
the order in which it has to be actually carried out, and in general 
following the course of the second part of the analysis, the resolution ; 
(2) the demonstration that the figure obtained does satisfy all the given 
conditions, which follows the steps of the first part of the analysis, 
the transformation, but in the reverse order. The second part of 
the analysis, the resolution, would be much facilitated and shortened 
by the existence of a systematic collection of Data such as Euclid's ^ 
book bearing that title, Jbonsisting of propositions proving that, if 
in a figure certain parts or relations arc given, other parts or relations 
are also given(/ As regards the first part of the analysis, the trans- 
formation, the usual rule applies that every step in the chain must 
be unconditionally convertible; and any failure to observe this 
condition will be brought to light by the subsequent synthesis. 
The second part, the resolution, can be directly turned into the 
construction since that only is given which can be constructed by 
the means provided in the Elements. 

It would be difllicult to find a better illustration of the above than 
the example chosen by Hankel from Pappus.^ 

Given a circle ABC and two joints D, E external to it, to draw 
straight lines DB, l£.Efrom D,E to a point B on the circle such that, 
if DB, IL^ produced meet the circle again in C, A, AC shcdl be parallel 
to DE. 

Analysis. 

Suppose the problem solved and the tangent at A drawn, meeting 
ED produced in F. 

(Part I. Transformation.) 

Then, since AC is parallel to D£, the angle at C is equal to the 
angle CDE. 

But, since FA is a tangent, the angle at C is equal to the angle FAE. 

Therefore the angle FAE is equal to the angle CDE, whence A, 
By D, F are concyclic. 

* 2^then, p. 93. • Hankel, p. 141. ' » Pappus, vii. pp. 830— «. 



143 INTRODUCTION 



I 

[CH. UL |6 I 




Therefore the rectangle AE^ EB is equal to tiie rectan^e FR^ 
ED. 

(Part II. Resolution.) 

But the rectangle AE^ EB is given, 
because it is equal to the square on the 
tangent from E. 

Therefore the rectangle FE^ ED is 
given; 

and, since ED is given, FE is nven (in 
length). [Data, 57J 

But FE is given in position also,- so 
that F is also given. [Daia^ 27.] 

Now FA is the tangent from a given point F to a circle ABC 
given in position ; 
therefore FA is given in position and magnitude. [Daia^ 9a] 

And F is given ; therefore A is given. 

But E is also given ; therefore the straight line AE is given in 
position. [Dttia^ 26.] 

And the circle ABC is given in position ; 
therefore the point B is also given. [Data^ 35.] 

But the points D, E are idso given ; 
therefore the straight lines DB^ BE are also given in position. 

Synthesis. 

(Part I. Construction.) 

Suppose the circle ABC and the points A E given. 

Take a rectangle contained by ED and by a certain strai^t 
line EF equal to the square on the tangent to the circle from E. 

From F draw FA touching the circle in A ; join ABE and then 
DB^ producing DB to meet the circle at C. Join A C. 

1 say then that AC is parallel to DE. 

(Part II. Demonstration.) 

Since, by hypothesis, the rectangle FEy ED is equal to the square 
on the tangent from Ey which again is equal to the rectangle AEy EB^ 
the rectangle AEy EB is equal to the rectangle FEy ED. 

Therefore AyByDyF are concyclic, 
whence the angle FAE is equal to the angle BDE. 

But the angle FAE is equal to the angle ACB in the alternate 
segment ; 
therefore the angle A CB is equal to the angle BDE. 

Therefore AC\s parallel to DE. 

In cases where a Siopurfio^ is necessary, i.e. where a solution is 
only possible under certain conditions, the analysis will enable those ] 
conditions to be ascertained. Sometimes the Siopiafio^ is stated and 
proved at the end of the analysis, e.g. in Archimedes, On the Sphere 
and Cylindery II. 7 ; sometimes it is stated in that place and the proof 
postponed till after the end of the synthesis, e.g. in the solution of 
the problem subsidiary to On tfte Sphere and Cylindery II. 4, preserved 
in Eutocius' commentary on that proposition. The analysis should 
also enable us to determine the number of solutions of which the 
problem is susceptible. 



' 



CH. ix.§7] THE DEFINITIONS 143 

§ 7. THE DEFINITIONS. 

General. " Real " and " Nominal " Definitions. 

It is necessary, says Aristotle, whenever any one treats of any 
whole subject, to divide the genus into its primary constituents, those 
which are indivisible in species respectively: e.g. number must be 
divided into triad and dyad ; then an attempt must be made in this 
way to obtain definitions, e.g. of a straight line, of a circle, and of 
a right angle'. 

The word for definition is 2/)09. The original meaning of this 
word seems to have been "boundary," "landmark." Then we have 
it in Plato and Aristotle in the sense of standard or determining 
principle ("id quo alicuius rei natura constituitur vel definitur," 
Index AristotelicusY ; and closely connected with this is the sense of 
definition. Aristotle uses both 8/309 and opurfio^ for definition, the 
former occurring more frequently in the Topics, the latter in the 
MetapAj^sics. 

Let us now first be clear as to what a definition does not do. 
There is nothing in connexion with definitions which Aristotle takes 
more pains to emphasise than that a definition asserts nothing as to 
the existence or non-existetice ol the thing defined. It is an answer 
to the question what a thing is (r/ ^cm), and does not say tliat it 
is (oTi itrrC). The existence of the various things defined has to be 
proved, except in the case of a few primary things in each science, 
the existence of which is indemonstrable and must be assumed among 
the first principles of each science ; c.g[. points and lines in geometry 
must be assumed to exist, but the existence of everything else must 
be proved. This is stated clearly in the long passage quoted above 
under First Principles'. It is reasserted in such passages as the 
following. "The (answer to the question) what is a man and the 
fact that a man exists are different things*.*' " It is clear that, even 
according to the view of definitions now current, those who define 
things do not prove that they exist'." "We say that it is by 
demonstration that we must show that everything exists, except 
essence (c* ft^ ovala tlri). But the existence of a thing is never 
essence; for the existent is not a genus. Therefore there must be 
demonstration that a thing exists. Thus, what is meant by triangle 
the geometer assumes, but that it exists he has to prove V "Anterior 
knowledge of two sorts is necessary : for it is necessary to presuppose, 
with regard to some things, that they exist \ in other cases it is 
necessary to understand what the thing described is, and in other 
cases It is necessary to do both. Thus, with the fact that one of two 
contradictories must be true, we must know that it exists (is true); 

» Anal, past 11. 13. b 15. 

* Cf. De aninia, 1. 1, 404 a 9, where ** breathing " is spoken of as the 6pot of ** life," and 
the many passages in the Politics where the wora is used to denole that which gives its 
special character to the several forms of government (virtue being the tpot of aristocracy, 
wealth of oligarchy, liberty of democracy, 1194 a 10) ; Plato, Republic^ viii. 551 c. 

' Anal. post. I. 10, 76 a 31 sqq. ^ ibid. Ii. 7, 91 b 10. 

* ibid. 91 b 19. * ibid. 91 b 11 sqq. . 



144 INTRODUCTION [CB. n. f 7 

of the triangle we must know that it means such and such a tiling ; of 
the unit we must know botii what it means and that it exists^" What 
is here so much insisted on is the very fact which Mill pointed out 
in his discussion of earlier views of Definitions, where he says that 
the so-called real definitions or definitions of tkin^^ do not constitute 
a different kind of definition from nominal definitions, or definitions 
of names ; the former is simply the latter pirns something else, namely 
a covert assertion that the thii^ defined exists. ''This covert assertion 
is not a definition but a postulate. The definition is a mere identical 
proposition which gives information only aboiit the use of language, 
and from which no conclusion affecting matters of fact can possibly 
be drawn. The accompanying postulate, on the other hand, affirms 
a fact which may lead to consequences of evenr degree of importance. 
It affirms the actual or possible existence of Things possessing the 
combination of attributes set forth in the definition : and this, if true, 
may be foundation sufficient on which to build a whole fabric of 
scientific truth'." This statement really adds nothing to Aristotie*s 
doctrine': it has even the slight disadvantage, due to the use of 
the word "postulate" to describe "the covert assertion* in all cases, 
of not definitely pointing out that there are cases where existence 
has to be proved as distinct from those where it must be assmmd. 
It is true that the existence of a definiend may have to be taken 
for granted provisionally until the time comes for proving it; but, 
so far as r^ards any case where existence must be proved sooner 
or later, the provisional assumption would be for Aristotie, not a 
postulate, but a hypothesis. In modem times, too. Mill's account of 
the true distinction between real and nominal definitions had been 
fully anticipated by Saccheri*, the editor of Euclides ab omni naroo 
vvtdicatus (1733), famous in the history of non-Euclidean geometry. 
In his Logica Demofistrativa (to which he also refers in his Euclid) 
Saccheri lays down the clear distinction between what he calls de- 
finitiones quid nominis or nominaleSy and definitiones quid rei or reales^ 
namely that the former are only intended to explain the meaning 

1 AfuU.posL I. I, 71 a II sqq. ' Mill's System 0/ Logic, Bk. i. ch. viii. 

< It is true that it was in opposition to ** the ideas of most of the ArisMelian hgidans** 
(rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his 
observation: ** We have already made, and shall often have to repeat, the remark, that the 
philosophers who overthrew Realism by no means got rid of the consequences of Realism, 
but retained long afterwards, in their own philosophy, numerous propositions which could 
only have a rational meaning as part of a Realistic system. It had oeen handed down from 
Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry 
is deduced from definitions. This, so long as a definition was considered to be a proposition 
• nnfokling the nature of the thing,* did well enough. But Hobbes followed and rejected 
utterly the notion that a definition declares the nature of the thing, or does anything but 
state the meaning of a name ; yet he continued to affirm as broadly as any of his predecessors 
that the d^oZ, frincipia, or original premisses of mathematics, and even of all science, are 
definitions ; producing the singular paradox that systems of scientific truth, nay, all truths 
whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of 
mankind concerning the signification of words.** ' Aristotle was guilty of no such paradox ; 
on the contrary, he exposed it as plainly as did Mill. 

^ This has been fuDy brought out in two papers by G. Vailati, La i§cria ArisMdica ddim 
dkAmitione (Rwista di FiUsofia e uienu amm\ 1003)1 and Di tm* opera dtmen tu a /a del 
P. Gereiamo Saccheri (** Logica DemonstraUva,*' 1(^7) (in Rivista FUiofica^ I903)- 



f 



r 



CH. ix,§7] THE DEFINITIONS 145 

that is to be attached to a given term, whereas the latter, besides 
declaring the meaning of a word, affirm at the same time the existence 
of the thing defined or, in geometry, the possibility of constructing it 
The definitio quid nominis becomes a definitio quid rei " by means of a 
postulate^ or when we come to the question whether the thing exists and 
it is answered affirmatively ^^ Definitiones quid nominis are in them- 
selves quite arbitrary, and neither require nor are capable of proof; 
they are merely provisional and are only intended to be turned as 
quickly as possible into definitiones quid rei, either (i) by means of 
a postulate in which it is asserted or conceded that what is defined 
exists or can be constructed, e.g. in the case of straight lines and 
circles, to which Euclid's first three postulates refer, or (2) by 
means of a demonstration reducing the construction of the figure 
defined to the successive carrying-out of a certain number of those 
elementary constructions, the possibility of which is postulated. Thus 
definitiones quid rei are in general obtained as the result of a series of 
demonstrations. Saccheri gives as an instance the construction of a 
square in Euclid I. 46. Suppose that it is objected that Euclid had 
no right to define a square, as he does at the beginning of the Book, 
when it was not certain that such a figure exists in nature; the 
objection, he says, could only have force if, before proving and making 
the construction, Euclid had assumed the aforesaid figure as given. 
That Euclid is not guilty of this error is clear from the fact that 
he never presupposes the existence of the square as defined until 
after i. 46. 

Confusion between the nominal and the real definition as thus de- 
scribed, i.e. the use of the former in demonstration before it has been 
turned into the latter by the necessary proof that the thing defined 
exists, is according to Saccheri one of the most fruitful sources of 
illusory demonstration, and the danger is greater in proportion to 
the "complexity" of the definition, i.e. the number and variety of 
the attributes belonging to the thing defined. For the greater is the 
possibility that there may be among the attributes some that are 
incompatible, i.e. the simultaneous presence of which in a given figure 
can be proved, by means of other postulates etc. forming part of the 
I basis of the science, to be impossible. 

The same thought is expressed by Leibniz also. " If," he says, 
" we give any definition, and it is not clear from it that the idea, which 
we ascribe to the thing, is possible, we cannot rely upon the demon- 
strations which we have derived from that definition, because, if that 
idea by chance involves a contradiction, it is possible that even con- 
I tradictories may be true of it at one and the same time, and thus our 
■ demonstrations will be useless. Whence it is clear that definitions 
I are not arbitrary. And this is a secret which is hardly sufficiently 
known'." Leibniz' favourite illustration was the " regular polyhedron 
with ten faces," the impossibility of which is not obvious at first sight. 

^ ** Definitio quid nominis nata est evadere definitio auid rei per postulaium vel dum 
j Yenitur ad quaestionem an est et respondetur affirmative. 
I * O^cuUs eifragmenis inJdits de Leibniz, Paris, Alcan, 1903, p- 431* Quoted by Vailati. 

! H. B. 10 



146 INTRODUCTION [CB. dl 1 7 




It need hardly be added that, speaking generally, Euclid's defini- 
tions, and his use of them, agree with the doctrine of Aristotle 
that the definitions themselves say nothing as to the existence of die 
things defined, but that the existence of each of them most be 
pro>^ or (in the case of the ** prindple^'*) assumed. In geometry, 
says Aristotle, the existence of points and lines only must be as* 
sumed, the existence of the rest being proved. Accordingly Euclid's 
first three postulates declare the possmility of constructing straight 
lines and circles (the only "lines except straight lines UMd in the 
Elements). Other thin^ are defined and afterwards constructed and 
prov«l to exist : e.g. in cook L, Def. 20, it is explained what is meant 
by an equilateral triangle ; then (L i ) it is proposed to construct it, 
and, when constructed, it is proved to agree with the definitioa 
When a square is defined (l. De£ 22), the question whether such a 
thing really exists is left open until, in 1. 46, it is proposed to construct 
it and, when constructed, it is proved to satisfy the definition^ 
Similarly with the right angle (L Def. 10, and L ii) and parallels 
(L De£ 23, and I. 27 — 29). The greatest care is taken to exclude 
mere presumption and imagination. The transition from the sub- 
jective definition of names to the objective definition of things is 
made, in geometry, by means of coHStmctions (the first principles of 
lyhich are postulated), as in other sciences it is made by means ctf . 
experience*. } 

Aristotle's requirements in a definition. 

We now come to the positive characteristics by which, according 
to Aristotle, scientific definitions must be marked. 

Firsts the different attributes in a definition, when taken separately, 
cover more than the notion defined, but the combination of them i 
does not Aristotle illustrates this by the " triad," into which enter 
the several notions of number, odd and prime, and the last " in both 
its two senses (a) of not being measured by any (other) number (ck 
M^ fierpelaOai dpi0/i£) and (A) of not being obtainable by adding 
numbers together" (o>9 fi)f avyxeurOai i( dpiSfA&v), a unit not being a 
number. Of these attributes some are present in all other odd 
numbers as well, while the last [primeness in the second sense] \ 
belongs also to the dyad, but in nothing but the triad are they aU 
present •." The fact can be equally well illustrated from geometry. 
Thus, e.g. into the definition of a square (Eucl. L, Def. 22) there enter j 
the several notions of figure, four-sided, equilateral, and right-angled, | 
each of which covers more than the notion into which aU enter ks 
attributes! 

Secondly, a definition must be expressed in terms of things which 
are prior to, and better known than, the things defined*. This is 

^ Trenddenbarg, EUmmta Le^ices Aristotdeai^ | 5a 

* Trendelenbufg, Erldutenmgm tu den EUfnentm der arisUtelisckgn Lagik^ 3 ed. p. io7. 
On constniction as proof of existence in ancient geometry cf. H. G. ZcaXlktskt Die getmeiriselU 
Construction als ** ExUtenubem^s ** $n der amtUen GeonutrU (in Matkanatisckg Annalen, 
47. Band). 

* Anai.past, u. 13, 96 a 33— b i. 
^ Trendelenbnig, ErUhtieruneent p. 108. * Tsfia vi. 4, 141 a 96 iqq. 



CH. IX.J7] THE DEFINITIONS 147 

clear, since the object of a definition is to give us knowledge of the 
thing defined, and it is by means of things prior and better known 
that we acquire fresh knowledge, as in the course of demonstrations. 
But the terms " prior " and " better known " are, as usual susceptible 
of two meanings; they may mean (i) absolutely or logically prior and 
better known, or (2) better known relatively to us. In the absolute 
sense, or from the standpoint of reason, a point is better known than 
a line, a line than a plane, and a plane than a solid, as also a unit is 
better known than number (for the unit is prior to, and the first 
principle of, any number). Similarly, in the absolute sense, a letter is 
prior to a syllable. But the case is sometimes different relatively to 
us ; for example, a solid is more easily realised by the senses than a 
plane, a plane than a line, and a line than a point. Hence, while it is 
more scientific to begin with the absolutely prior, it may, perhaps, be 
permissible, in case the learner is not capable of following the scientific 
order, to explain things by means of what is more intelligible to him, 
'^ Among the definitions framed on this principle are £ose of the 
point, the line and the plane; all these explain what is prior by 
means of what is posterior, for the point is described as the extremity 
of a line, the line of a plane, the plane of a solid." But, if it is asserted 
that such definitions by means of things which are more intelligible 
relatively only to a particular individual are really definitions, it will 
follow that there may be many definitions of the same thing, one for 
each individual for whom a thing is being defined, and even different 
definitions for one and the same individual at different times, since at 
first sensible objects are more intelligible, while to a better trained 
mind they become less so. It follows therefore that a thing should 
be defined by means of the absolutely prior and not the relatively 
prior, in order that there may be one sole and immutable definition. 
This is further enforced by reference to the requirement that a good 
definition must state the genus and the differentiae, for these are 
among the things which are, in the absolute sense, better known than, 
and prior to, the species (j&v cEttXcS^ yvtoptiumipcuv koX irporipcav rov 
tXSav^ iarlp). For to destroy the genus and the differentia is to 
destroy the species, so that the former are ^ior to the species ; they 
are also better known, for, when the species is known, the genus and 
the differentia must necessarily be known also, e.g. he who knows 
** man " must also know " animal " and '' land-animal," but it does not 
follow, when the genus and differentia are known, that the species is 
known too, and hence the species is less known than they are^ It 
may be frankly admitted that the scientific definition will require 
superior mental powers for its apprehension ; and the extent of its 
use must be a matter of discretion. So far Aristotle ; and we have 
here the best possible explanation why Euclid supplemented his 
definition of a point by the statement in I. Def 3 that the extremities of 
a line are points and his definition of a surface by I. Def. 6 to the effect 
that the extremities of a surface are lines. The supplementary expla- 

* Topics VI. 4, 141 b «5— 34. 



148 INTRODUCTION [cH. iz. 1 7 

nations do in fact enable us to arrive at a better understanding of the 
formal definitions of a point and a line respectively, as is well ex* 
plained by Simson in his note on Def. I. Simson says» namelv. that 
we must consider a solid, that is, a magnitude wUch has length, 
breadth and thickness, in order to understand aright the definitions of 
a point, a line and a surfSsice. Consider, for instance, the boundary 
common to two solids which are contiguous or the boundary whidi 
divides one solid into two contiguous parts; this boundary is a surface. 
We can prove that it has no thickness by taking away either solid, 
when it remains the boundary of the other; for, if it had thickness, the 
thickness must either be a part of one solid or of the other, in which 
case to take away one or otiier solid would take away the thickness 
and therefore the boundary itself: which is impossible. Thereft)re 
the boundary or the surface has no thickness. In exactiy the same 
way, r^arding a line as the boundary of two contiguous surfaces^ we 
prove that the line has no breadth ; and, lastiy, re^urding a point as 
the common boundary or extremity of two lines, we prove that a 
point has no length, breadth or thickness. 

Aristotie on unscientific definitions. 

AristoUe distinguishes three kinds of definition which are un- 
scientific because founded on what b n^/ prior (^^ Ic wfHnipm(% The 
first is a definition of a 'thin|^ by means of its opposite, eg. of "* good ** 
by means of " bad " ; this is wrong because opposites are naUirally i 
evolved together, and the knowledge of opposites is not uiicommonly I 
regarded as one and the same, so that one of the two opposites '■ 
cannot be better known than tiie other. It is true that, m some 
cases of opposites, it would appear that no other sort of definition is 
possible: e.g. it would seem impossible to define double apart from the 
half and, generally, this would be the case with things which in their 
very nature (jcaO' aura) are relative terms {irpi^ n X^ctoa), since one 
cannot be known without the other, so that in the notion of either the 
other must be comprised as welP. The second kind of definition 
which is based on what is not prior is that in which there is a 
complete circle through the unconscious use in the definition itself of 
the notion to be defined though not of the name*. Trendelenburg 
illustrates this by two current definitions, (i) that of magnitude as 
that which can be increased or diminished, which is bad because the 
positive and negative comparatives "more" and "less" presuppose . 
the notion of the positive " great," (2) the famous Euclidean definition ^ 
of a straight line as that which "lies evenly with the points on itself" 
(^f laov roU t4^ iavrvf^ ayfieioi^ K€iT(u\ where " lies evenly " can only ^ 
be understood with the aid of the very notion of a straight line which is 
to be defined^ The tkird kind of vicious definition from that which 
is not prior is the definition of one of two coordinate species by means 
of its coordinate {dvriSinpfffiivov), e.g. a definition of " odd " as that 
which exceeds the even by a unit (the second alternative in Eud. vii. 
Def. 7) ; for "odd " and "even " are coordinates, being differentiae of 

^ Topia VI. 4, 143 A 13 — 31. * ihid, 143 a 34— b 6. 

'* Trendelenbuig, BrULwUntngtH^ P* I'S* 



_^ ) 



1 

I 

i CH. IX. § 7] THE DEFINITIONS 149 

number^ This third kind is similar to the first. Thus, says Tren- 
■f delenburg, it would be wrong to define a square as *'a rectangle 
\ with equal sides." 

Aristotle's third requirement. 
' A third general observation of Aristotle which is specially relevant 
' to geometrical definitions is that "to know what a thing is (rt ifrriv) is 
the same as knowing why it is (S^ rl iffrip)*.'* " What is an eclipse ? 
A deprivation of light from the moon through the interposition of the 
earth. Why does an eclipse take place? Or why is the moon 
eclipsed ? Because the light fails through the earth obstructing it 
What is harmony ? A ratio of numbers in high or low pitch. Why 
does the high-pitched harmonise with the low-pitched? Because 
the high and the low have a numerical ratio to one another*." *' We 
seek die cause (to hUtn) when we are already in possession of the 
fact {to Sri). Sometimes they both become evident at the same time, 
but at all events the cause cannot possibly be known [as a cause] 
before the fact is known^'* '' It is impossible to know what a thing is 
if we do not know that it is*/* Trendelenburg paraphrases : " The 
definition of the notion does not fulfil its purpose until it is made 
genetic. It is the producing cause which first reveals the essence of 
the thing. ••. The nominal definitions of geometry have only a 
provisional significance and are superseded as soon as they are made 
genetic by means of construction." Kg. the genetic definition of a 
parallelogram is evolved from Eucl. I. 31 (giving the construction for 
parallels) and I. 33 about the lines joining corresponding ends of two 
straight lines parallel and equal in length. Where existence is proved 
by construction, the cause and the fact appear together^ 

Again, *'it is not enough that the defining statement should set 
forth the fact, as most definitions do; it should also contain and 
present the cause ; whereas in practice what is stated in the definition 
is usually no more than a conclusion (avfiiripaa^). For example, 
{I what is quadrature ? The construction of an equilateral right-angled 
•I figure equal to an oblong. But such a definition expresses merely the 
f conclusion. Whereas, if you say that quadrature is the discovery of a 
i mean proportional, then you state the reason ^" This is better under- 
i stood if wc compare the statement elsewhere that "the cause is the 
'middle term, and this is what is sought in all cases*," and the illustra- 
' tion of this by the case of the proposition that the angle in a semi- 
circle is a right angle. Here the middle term which it is sought to 
establish by means of the figure is that the angle in the semi-circle is 
equal to the half of two right angles. We have then the syllogism : 
Whatever is half of two right angles is a right angle ; the angle in a 
i semi-circle is the half of two right angles ; therefore {conclusion) the 
-angle in a semi-circle is a right angle*. As with the demonstration, so 

* Topia VI. 4, 143 b 7—10. — • Anal, past, 11. 3, 90 a 31. 

* Anal. post. 11. 3, 90 a 15—31. — * ibid, ii. 8, 93 a 17. 

* ihid, 9^ a 3a * Trendelenbarg, Erldutemngm, p. 116. 
' Dt amma 11. 3, 413 a 13 — 30. * Anal* post. \\. 3, 90 a 6, 

* Hid. II. II, 94 a 33. 



1 



ISO INTRODUCTION [cB.ix.|7 

it should be with the definition. A definition idiich is to show the 
genesis of the thing defined should contain the middle term or cause ; 
otherwise it is a mere statement of a conclusion. Consider, for 
instance, the definition of "quadrature" as ''making a square equal in 
area to a rectangle with unequal sides." This gnres no hint as to 
whether a solution of the problem is possible or how it is solved : bnt, 
if you add that to find the mean proportional between two given 
straight lines gives another straight une such that the square on it is | 
equal to the rectangle contained by the first two straight linesi yoa ■ 
supply the necessary middle term or cause^. ! 

Technical terms not defined by Euclid. | 

It will be observed that what is here defined, ''quadrature* or 
" squaring " (rerpaywviafAosi), is not a geometrical figure, oran attribute 
of such a figure or a part of a figure, but a tedmical term used to , 
describe a certain problem. Euclid does not define such things ; but 
the fact that Aristotle alludes to this particular definition aswdl as to 
definitions of deflection {tc€tc\M0ai) and of verging' {vwlkuf) seems to 
show that earlier text-books included among ddinitions explanations 
of a number of technical terms, and that Euclid deliberat^ omitted 
these explanations from his Elements as surplusage. LaCer the 
tendency was again in the opposite direction, as we see from the mudi 
expanded Definitions of Heron, which, for example, actually include 
a definition of a deflected line (xeKkaafUpff ypa^Lia^nr* Eudid uses the 
passive of kX&p occasionally*, but evidently considered it unneoesraiy 
to explain such terms, which had come to bear a recognised meaning. 
The mention too by Aristotle of a definition of verging (yci/tiy) 
suggests that the problems indicated fa^ this term were not excluded 
from elementary text-books before Euclid. The type of problem 
(vevcisi) was that of placing a straight line across two lines, e.g. two 
straight lines, or a straight line and a circle, so that it shall verge to a 
given point (i.e. pass through it if produced) and at the same time the 
intercept on it made by the two given lines shall be of given length. 



Other posages in Aristotle may be quoted to the like effect : e.g. Anai, pott. i. t, 
71 b 9 '* We consider that we know a particniar thing in the absolate sense, as distinct 
nrom the sophistical and incidental sense, when we consider that we know the cause on 
aooonnt of which the thing is, in the sense of knowing that it is the canse of that thing and 
that it cannot be otherwise,*' ibid. i. 13, ^pa 1 '* For here to know ihtfact is the fnnctum of! 
those who are concerned with sensible thm^ to know the cauu is the ranction of the matbe- , 
matician ; it is he who possesses the proofi of the causes, and often he does not know the 
fiict." In view of such passages it is difficult to see how Proclus came to write (p. sos, 11) 
that Aristotle was the originator ('A^c^rorAow Korif^pmn) of the idea of Amphmomus ana 
others that geometry does not investigate the cause and the wfy {rh Ml W)- To this Geminus 
replied that the invertigation of the cause does, on the contrary, appear in geometry. " For 
how can it be maintained that it is not the business of the geometer to inauire for wluit reaaon, 
on the one hand, an infinite number of equilateral polygons are inscribea in a circle, but, on 
the other hand, it is not possible to inscribe in a sphere an infinite numbo' of polyhedral 
figures, ejquilateral, equiangular, and made up of similar plane figures? Whose business is it 
to ask this question and find the answer to it if it is not that of the geometer? Now when 
geometers reason per imfossibiU thejr are content to discover the property, but when they , 
argue by direct proof, it such pool be only partial (M Mpovf)i thb does not suffice for 
showing the canse ; if however it is general and applies to all like cases, the why (vt^ M W) 
is at once and concurrently made evident." 

* Heron, ed. Hultsch, Def. 14, p. 11. * e.g. in III. 90 and in IkOa 89. 



IcH. IX.J7] THE DEFINITIONS 151 

In general, the use of conies is required for the theoretical solution of 
I these problems, or a mechanical contrivance for their practical 
j| solution \ Zeuthen, following Oppermann, gives reasons for supposing, 
■ not only that mechanical constructions were practically used by the 
f older Greek geometers for solving these problems, but that they were 
theoretically recognised as a permissible means of solution when the 
solution could not be effected by means of the straight line and circle, 
and that it was only in later times that it was considered necessary to 
use conies in every case where that was possible*. Heiberg* suggests 
that the allusion of Aristotle to yevo-ew perhaps confirms this sup- 
position, as Aristotle nowhere shows the slightest acquaintance with 
conies. I doubt whether this is a safe inference, since the problems 
of this type included in the elementary text-books might easily have 
been limited to those which could be solved by " plane " methods (i.e. 
by means of the straight line and circle). We know, e.g., from Pappus 
that Apollonius wrote two Books on plane vevaei^*. But one thing 
is certain, namely that Euclid deliberately excluded this class of 
Iproblem, doubtless as not being essential in a book of Elements. 
Definitions not afterwards used. 

Lastly, Euclid has definitions of some terms which he never after- 
wards uses, e.g. oblong (irepo/AffKe^), rhombus, rhomboid, trapezium. 
The "oblong" occurs in Aristotle; and it is certain that all these 
definitions are survivals from earlier books of Elements. 

^ Cf. the chapter on vw&vtit in ne IVarks of Archimedes^ pp. c — cxxii. 

* Zeuthen, Die Lehre twn den Kegelschniiten im Altertum^ ch. ii, p. 161. 
' Heiberg, Mathemaiiseka tu ArisioieUs^ p. 16. 

* Pi^pns VII. pp. 670—3. 



1 



,>■■/ 



BOOK I. 

DEFINITIONS. 

•^ I. A point is that which has no part. 
<- 2. A line is breadthless length. 

3. The extremities of a line are points. 

— 4. A straight line is a line which lies evenly with the 
points on itself. 

— 5* A surface is that which has length and breadth only. 

6. The extremities of a surface are lines. 

7. A plane surface is a surface which lies evenly with 
the straight lines on itself. 

8. A plane angle is the inclination to one another of 
two lines in a plane which meet one another and do not lie in 
a straight line. 

9. And when the lines containing the angle are straight, 
the angle is called rectilineal. 

10. When a straight line set up on a straight line makes 
the adjacent angles equal to one another, each of the equal 
anfi^les is right, and the straight line standing on the other is 
called a perpendicular to that on which it stands. 

11. An obtuse angle is an angle greater than a right 
angle. 

12. J^ii^iacute angle is an angle less than a right angle. 
— 13. A boundary is that which is an extremity of any- 
thing. 

■^ 14. A figure is* mat which js contained by any boundary 
or boundaries. 

15. A circle is a plane figure contained by one line such 
that all the straight lines falling upon it from one point among 
those lying within the figure are equal to one another ; 



154 BOOK I [h VIEW. i6— POST. 4 

16. And the point is (Ailed the centre of the circle. 

1 7. A diameter of the circle is any straight line drawn 
through the centre and terminated in both directions by the 
circumference of the circle, and such a straight line also 
bisects the circle. 

1 8. A semicircle is the figure contained by the diameter 
and the circumference cut off by it . And the centre of the 
semicircle is the same as that of the circle. 

19. Rectilineal figures are those which are contained 
by straight lines, trilateral figures being those contained by 
three, quadrilateral those contained j^ four, and multi- 
lateral those contained by more than four straight lines. 

20. Of trilateral figures, an equilateral triangle is that 
which has its three sides equal, an isosceles triangle that 
which has two of its sides alone equal, and a scalene 
triangle that which has its three sides unequal. 

21. Further, of trilateral figures, a right-angled tri- 
angle is that which has a right angle, an obtuse-angled 
triangle that which has an obtuse angle, and an acute- 
angled triangle that which has its three angles acute. 

22. Of quadrilateral figures, a square is that which is 
both equilateral and right-angled ; an oblong that which is 
right-angled but not equilateral ; a rhombus that which is 
equilateral but not right-angled ; and a rhomboid that which 
has its opposite sides and angles equal to one another but is 
neither equilateral nor right-angled. And let quadrilaterals 
other than these be called trapezia. 

23. Parallel straight lines are straight lines which, 
being in the same plane and being produced indefinitely in 
both directions, do not meet one another in either direction. 



POSTULATES. 

Let the following be postulated : 

1. To draw a straight line from any point to any point 

2. To produce a finite straight line continuously in a 
straight line. 

3. To describe a circle with any centre and distance. 

4. That all right angles are equal to one another. 



-L«. . > 



^ 






I- POST. 5— c N. s] DEFINITIONS ETC 155 

5* . That, if a straight line falling on two straight lines 
make the interior angles on the same side less than two right 
angles, the two straight lines, if produced indefinitely, meet 
on that side on which are the angles less than the two right 
angles. 

COMMON NOTIONS. 

1. Things which are equal to the same thing are also 
equal to one another. 

2. If equals be added to equals, the wholes are equal. 

3. If equals be subtracted from equals, the remainders 
are equal. 

[7] 4- Things which coincide with one another are equal to 

one another. 

[8] 5. The whole is greater than the part 

Definitio^i I. 

iilfUUv loTiF, oS fUpot aiOiv, 
'^'^ A point is that wAick has no part. 

An exactly parallel use of lUfioi {i<rrC) in the sin^lar is found in Aristotle, 
Metaph. 1035 b 32 fiipoi fuv cZv Icrrl fcou rov cZSou^ literally *'There is a 
/jTf/ even of the form "; Bonitz translates as if the plural were used, *'Theile 
giebt es," and the meaning is simply "even the form is divisiMe (into parts)." 
Accordingly it would be quite justifiable to translate in this case "A point is 
that which is indivisible intoparts.^ 

Martianus Qipella (5th c. a.d.) alone or almost alone translated differently, 
**Punctum est cuius pars nihil t&\^^ *'a point is that a part of which is nothing,^ 
Notwithstanding that Max Simon {Eudid und die sechs planimitrischen Bilcher^ 
190 1 ) has adopted this translation (on grounds which I shall presently mention), 
I cannot think that it p^ives any sense. If a part of a point is nothings Euclid 
Might as well have said that a point is itself ''nothing,'' which of course he 
does not do. 

Pre-Euclidean definitions. 

It would appear that this was not the definition given in earlier text- 
books; for Aristotle {Topics vi. 4, 141 b 20), in speaking of *7A^ definitions'' 
of point, line, atod surface, says that they all define the prior by means of the 
posterior, a point as an extremity of a line, a line of a surface, and a surface 
of a solid. 

The first definition of a point of which we hear is that given by the 
Pythagoreans (cf. Proclus^ o. 95, ai), who defined it as a "monad having 
position" or ''with position added" (iiova^ wfHxrXaPovau O&riv). It is firequently 
wed by Aristotle, either in this exact form (cf. De anima L-4^ 409 a 6) or its 
equivalent: e.g. in Metaph, 7016 b 24 he says that that which is indivisible 
•very way in respect of ma^ itude and qu& magnitude but has not position is 
^wumady while Uiat which is similarly indivisible and has position is 9. paint. 

Plato ippMR ID IVPO obfected to this definition. Aristotle says (Meti^h. 



156 BOOK I [l 

993 a 3o) that he objected "to tfab genua [that of pomti]a$ being a geometrical 
fiction {yu^jutrpudw Uy/ia)^ and caUed a point the beginning m a line (4«^ 
ypa/ifi'^), while again he fre(}uently npoke of 'indiTiaUe lines."* To wbidi 
Aristotle replies that even ''indivisil^ lines" most have eitreoiities, so that 
the same a^ment which proves the existence of Ufus can be used to prove 
ataX points exist It would appear therefore that, when Aristotle objects to 
the definition of a point as die extremity of a Une (v^pot ypo^i^) as un- 
scientific {Topics VI. 4, 141 b 21X he is aiming at Plato. Heiberg conjectures 
{MathemaHsches mu Aristotdes^ p. 8) diat it was due to Plato's influoice that 
the word for "point" generally used by Aristode (<myfu|) was r^laoed by 
tnuMwv (the r^iular term used by Eudid, Archimedes and later wnters), tfie 
latter term {^nota^ a conventional mark) probably hmg conddeied more 
suitable than frrvfi»alj (fLpunchtre) whidi mij^ appear to ^lim greater naUiy 
for a point 

Anstode's conception of a point as diat which is indivisible and has 
position is fiirther illustrated by such observations as diat a point is not a 
body (De caeh 11. 13, 296 a 17) and has no wri^ {ibiJL in. i, S99 a 30); 
again, we can make no distinction between a point and the/Auy (rmf) where 
it is {Physics iv. i, 209 a 1 1). He finds the usual difficulty in aooountiii^ for 
the transition fix>m the indivisible, or infinitely small^ to the finite or divisible 
magnitude. A point being indivisiUe^ no accumulation of points, however fitf 
it mav be carried, can give us anything divisiUe, whereas of course a line is a 
divisible ma^tude. Hence he holds that points cannot make up anythiiig 
continuous like a line, point cannot be continuous with point (00 yap «Eot&v 
ixi/uyov arifUiov arifuicv ^ oriyfuf any§KlJ9t -Di gtm» d corr. I. a, 317 a lo), and 
a line is not made up of points (ov myMtrw cic frttfium^ Pkysia nr. 8, ai^ 
b 19). A point, he says, is like the mm in time : nam is indivisiUe and is 
not 9k part of time, it is only the b^;inning or end, or a division, of time, and 
simikurly a point may be an extremity, b^inning or division of a line, but is 
not part of it or of magnitude (cf. De cado in. i, 300 a 14, Pf^sia iv. zi, 
220 a I — 21, VI. X, 231 b 6 sqq.). It is only by motion that a point can 
generate a line {De anima i. 4, 409 a 4) and thus be the origin of magnitude. 

Other ancient definitions. 

According to an-NauIzi (ed Curtze, p. 3) one "Herundes" (not so fiu: 
identified) defined a point as "the indivisible beginning of all magnitudes," 
• and Posidonius as ''an extremity which has no dimension, or an extremity of 
aline." 

Criticisms by commentators. 

Euclid's definition itself is of course practically the same as that which 
Aristotle's fi'equent aUusions show to have been then current, except that it 
omits to say that the point must have position. Is it then sufficient, seeing 
that there are other things which are without parts or indivisible, e.g. the now 
in time, and the unit in number? Proclus answers (p. 93, 18) that the point 
is the only thing in the subject-matter of geometry that is indivisible. Relatively 
therefore to the particular science the definition is sufficient Secondly^ the 
definition has been over and over again criticised because it is purely n^;ative. 
Proclus* answer to this is (p. 94, 10) that nq;ative descriptions are appropriate 
to first principles, and he quotes Pannenide s as having described his first and 
last cause by means of n^;ations merely. Arif totie too admits that it may 
sometimes be necessary for one framing a definition to use negations, e.g. in 
defining privative terms such as "blind"; and he seems to accept as proper 



■I 



L DEF. i] NOTE ON DEFINITION i 157 

the negative element in the definition of a point, since he says {De anima iii. 6, 
430 b 20) that ''the point and every division [e.g. in a length or in a period 
of time], and that which is indivisible in this sense, is exhibited as privation 

Simplicius (quoted by an-NairIzi) says that ''a point is the beginning of 
magnitudes and that from which they grow ; it is also the only thing wluch, 
having position, is not divisible.'' He, like Aristotle, adds that it is by its 
motion that a point can generate a magnitude : the particular magnitude can 
only be "of one dimension," viz. a line, since the point does not "spread 
itself" (dimittat). Simplicius further observes that Euclid defined a point 
n^atively because it was arrived at by detaching surface from body, line from 
surface, and finally point from line. "Since then body has three dimensions 

I It follows that a point [arrived at after successively eliminating all three 
dimensions] has fwne of the dimensions^ and has no part." This of course 
reappears m modern treatises (cf. Rausenbeiger, Eiementar-geometrie des 
Punktes^ der Geraden und der Ebtne^ 1S87, p. 7). 

An-NairizI adds an interesting observation. " If any one seeks to know 
the essence of a point, a thing more simple than a line, let him, in the sensible 
world, think of the centre of the universe and the poUs,^^ But there is 
nothing new under the sun : the same idea is mentioned, in an Aristotelian 
treatise, in controverting those who imagine that the poles have some influence 
in the motion of the sphere, "when the poles have no magnitude but are 
extremities and points" (De motu animalium 3, 699 a 21). 

Modem views. 

In the new geometry represented by the excellent treatises which start 
from new systems of postulates or axioms, the result of the profound study of 
the fundamental principles of geometry during recent years (I need only 
mention the names of Pasch, Veronese, Enriques and Hilbert), points come 
before lines^ but the vain effort to define them a priori is not made ; instead 
of this, the nearest material things in nature are mentioned as illustrations, 
with the remark that it is from them that we can get the abstract idea. Cf. 
the full statement as regards the notion of a point in Weber and Wellstein, 
Encyclopddit der elementaren Mathematik^ 11., 1905, p. 9. "This notion is 
evolved from the notion of the real or supposed material point by the process 
of limits, Le. by an act of the mind which sets a term to a series of presen- 
tations in itself unlimited. Suppose a grain of sand or a mote in a sunbeam, 
which continually becomes smaller and smaller. In this way vanishes more 
and more the possibility of determining still smaller atoms in the grain of 
sand, and there is evolved, so we say, with growing certainty, the presentation 
of the point as a definite position in space which is one and is incapable of 
further division. But this view is untenable ; we have, it is true, some idea 
how the grain of sand gets smaller and smaller, but only so long as it remains 
just visible; after that we are completely in the dark, and we cannot see or 
imagine the further diminutiop. That this procedure comes to an end is 
unthinkable ; that nevertheless diere exists a term beyond which it cannot go, 
we must believe or postulate without ever reaching it . . . It is a pure 
act of wili^ not of the understanding." Max Simon observes similarly (Euclid^ 
p. 25) " The notion * point ' belongs to the limit-notions (Grenzbegriffe), the 
necessary conclusions of continued, and in themselves unlimited, series of 
presentations." He adds, "The point is the limit of localisation; if this is 
more and more energetically continued, it leads to the limit-notion 'point,' 



158^ BOOK I ' [i. DiTP. I, s 

better 'position,' which at the tame time uivohFa a chai^ Content 

of space vanishes, relative pasiUm remains. 'Pdnt' theny aocoiding to our 
interpretation of Euclid, is the eztremest limit of diat iriiidi we can £il think 
of (not observe) as a i;^^/^ presentation, and if we go further than that^ not 
only does extension cease but even relative /Auy^ and in this lenie tfie 'put' 
is nothing.*^ I confess I think that even the meanii^ which Simon intend! to 
convey is better expressed by ''it has 110 part" than uf "the part is nothing,* 
since to take a ''part" of a thing in Eudid's tense ii the result of a simple 
division, corresponding to an arithmetical fraction, would not be to change 
the notion from that of the thing divided to an entirely diflTerent < 



f 



Definition 2. 

TpofLfi^ Sk firJKOi airXarcc 

A line is breadihUss length. 

This definition may safely be attributed to the Platonic School, if not to 
Plato himself. Aristotle {Topies vl 6, 143 b 11) speaks of it as open to 
objection because it " divides the goius by n^^ation," loigth being neonsaiily 
either breadthless or possessed ofbfeadUi; it woukl seem however that tfie 
objection was only taken in order to score a point against the Phtonists, ainoe 
he says {ibid. 143 b 29) that the aigument is "of service mtfy against dioae 
who assert that the genus [sc. length! is one numerically, diat is^ thoae who 
assume ideas^^* e.g. &e idea of lengta (oAri ^i^Kot) which they r^^ard as a 
genus : for if the genus, being one and self-existent, could be divided into 
two species, one of which asserts what the other denies, it would be sdf- 
contradictory (Waitz). 

-Proclus (pp. 96, 21 — 97, 3) observes that, whereas the definition of a point 
is merely ne^tive, the line introduces the first " dimension," and so its 
definition is to this extent positive^ while it has also a n^;ative element which 
denies to it the other "dimensions" (Suurrao-ctf). The negation of both 
breadth and depth is involved in the single expression "breadthless" (onrXarcg), 
since everything that is without breadth is also destitute of depth, though the 
converse is of course not true. 

Alternative definitions. 

The alternative definition alluded to by Produs, ficycfct c^* tr tMurrftror 
" magnitude in one dimension " or, better perhaps, " magnitude extended one 
way " (since fiuurrao-ic as used with reference to line, sur&ce and solid scarcdy 
corresponds to our use of "dimension" when we speak of "one,'' "two," or 
"three dimensions"), is attributed by an-NairIzi to "Heromides/* who must 
presumably be the same as "Herundes," to whom he attributes a certain 
definition of a point It appears however in substance in Aristotie, though 
AristoUe does not use the adjective Suurrardp, nor does he apparenUy use 
Suunroo-if except of body as having three " dimensions " or " having dimension 
(or extension^ oi/ways (vdb^)," the "dimensions" being in his view (i) up 
and down, (2) before and behind, and (3) right and left, and "up" being the 
principle or beginning of lengthy *< ri^ht " of breadth^ and " before " of d^th 
(De caelo ii. 2, 284 b 24). A line is, according to Aristotle, a magnitude 
^divisible in one way only" (fun^xv ^Mupcro^X ^ contrast to a ma^tude 
divisible in two ways (&X9 ficoi^ror), or a surface, and a magnitude divisible 
"in all or in three wa^s" {^riirqi ttal rpi)^ iiaip€r6y\ or a body {Meiaph. 
1016 b 25 — 27); or It is a magnitude **cmtintkms one way (or in one 
directionX" as compared with magnitudes continuous two ways or thne ways. 



^ 



I. DEF. a] NOTES ON DEFINITIONS i, a 159 

which curiously enough he describes as ** breadth " and "depth" respectively 
(/Acyctfos 8^ rh fuv i^t* tv ovKcxcf fi'^KOSf rh 8* ^l Suo irX<£ro9, ro 8* ^i rpia piBo^, 
Metaph, loao a 11), though he immediately adds that "length " means a line, 
" breadth " a surface, and " depth " a body. 

Proclus gives another alternative definition as ^^flux of a point '^ (fiwrn 
fnujuiiov), i.e. the path of a point when moved. This idea is also alluded to in 
Anstotle (De anima i. 4, 409 a 4 above quoted) : " they say that a line by its 
motion produces a surface, and a point by its motion a line." "This 
definition," says Proclus (p. 97, 8 — 13), "is a perfect one as showing the 
essence of the line : he who called it the flux of a point seems to define it 
from its genetic cause, and it is not every line that he sets before us, but only 
the immaterial line ; for it is this that is produced by the point, which, though 
itself indivisible, is the cause of the existence of things divisible." 

Proclus (p. 100, 5 — 19) adds the usefiil remark, which, he says, was 
current in the school of ApoUonius, that we have the notion of a line when we 
ask for the length of a road or a wall measured merely as length ; for in that 
case we mean something irrespective of breadth, viz. distance in one 
" dimension." Further we can obtain sensible perception of a line if we look 
at the division between the light and the dark when a shadow is thrown on 
the earth or the moon ; for clearly the division is without breadth, but has 
length. 

Species of ** lines." 

After defining the "line" Euclid only mentions one species of line, the 
straight line, although of course another species appears in the definition of a 
circle later. He doubtless omitted all classification of lines as unnecessary for 
his purpose, whereas, for example. Heron follows up his definition of a line by 
a division of lines into (i) those which are " straight " and (2) those which are 
not, and a further division of the latter into {a) "circular circumferences," 
(b) "spiral-shaped" (IXucofiScis) lines and {c) "curved" (fcofiirvAat) lines generally, 
and then explains the four terms. Aristotle tells us {Metaph. 986 a 25) that 
the Pythagoreans distinguished straight (cv^ and curved (fcofiirvAov), and this 
distinction appears in Plato (cf. Republic x. 602 q) and in Aristotle (cf. " to a 
line belong tiie attributes straight or curved," AnaL post. i. 4i 73 b 19; "as in 
mathematics it is useful to know what is meant by the terms straight and 
curved," De anima i. i, 402 b 19). But firom the class of "curved" lines 
Plato and Aristotle separate off the vcpi^cpiTs or " circular " as a distinct 
species often similarly contrasted with straight Aristotle seems to recognise 
broken lines forming an angle as one line : thus "a line, if it be bent (kcko/a- 
ficn7), but yet continuous, is called one" {Afetaph, 1 01 6 a 2); "the straight line 
is more one than the bent line" (ibid. 1016 a 12^. Cf. Heron, Def. 14, "A 
broken line («cc«cXa<rficn7 ypofifiif) so-called is a luie which, when produced, 
does not meet itse^.** 

When Proclus says that both Plato and Aristotle divided lines into those 
which are "straight," "circular" (ircpi^c/n;^) or "a mixture of the two," adding, 
as regards Plato, that he included in the last of these classes " those which are 
caUed helicoidal among plane (curves) and (curves) formed about solids, and 
such species of curved lines as arise from sections of solids" (p. 104, i — 5), 
he appears to be not quite exact The reference as regards Plato seems to be 
to Parmenides 145 b: "At that rate it would seem that the one must have 
shape, either straight or round (arrpoyyuXov) or some combination of the two"; 
but this scarcely amounts to a formal classification of lines. As regards 



i6o 



BOOK I 



[h i«r. t 



Aristotle, Produs seems to have in mind the pasage (Ik mcfr i. 9» t68 b i^) 
where it is stated that ''all maiiom in qpace, which we call translatioa (4^p£^ is 
(in) a straight line, a circle, or a combmation of the two; for the first two are 
the only simple (motions)** 

For completeness it is desirable to add the substance of Produs* account 
of the classification of lines, for frtiich he quotes Geminus as his authority. 

Geminus' first classification of lines. 

This bqi;ins (p. 1 1 1, i — o) with a division of lines into tmt^osik (otMctoc) 
and incomposite (dirvi^cTot). The only illustration given of the iomforiU 
class is the *' broken line which forms an angle** (^ «cicXa<r/&^ rai yn way 
voiovo-a) ; the subdivision of the huomposiii class then follows (in the test as 
it stands the word " composite " is clearly an error for *' incomposite % The 
subdivisions of the incompcMite class are repeated in a later passage (pp- 176, 
27 — 177, 23) with some additional details. The following diagram reproduces 
the effect of both versions as fiu: as possible (all the illustrations mentioned Iqr 
Produs being shown in brackets). 

Unet 



oonipotitc iooonipotite 

(broken line fbnmng an ana^) I 



forming a figure 

or determinate 

(drde, ellipse, dssoid) 



not forming a fignie 

or 
indeCenninate 



extending witboot limit 
(itraia;ht Une, parabola, nypeibola. 



I) 



The additional details in the second version, which cannot easily be shown 
in the diagram, are as follows : 

(i) Of the lines which extend without limit, some do not/arm a figure at 
aU (viz. the straight line, the parabola and the hyperbola); but some first 
*'come together and form a figure ** (Le. have a loop), ''and, for the rest, 
extend witibout limit" (p. 177, 8). 




As the only other curve, besides the parabola and the hyperbola, which 
has been mentioned as proceeding to infinity is the conchoid (of Nicomedes), 
we can hardly avoid the condusion of Tannery^ that the curve which has a 
loop and then proceeds to infinity is a variety of the conchoid itself. As is 

^ Noiapomr tkistoire dis lignes it surf oca ccmrbes dans ftuUiquHi in BnUetim dts sciem^ 
mathhn, dastronom. t i6r. viii. (1884), PP* '08— 9. 



■j 



L DEF. a] NOTE ON DEFINITION a x6i 

well known, the ordinary conchoid (which was used both for doubling the 
cube and for trisecting the angle) is obtained in this way. Suppose any 
number of rays passing through a fixed point (the pole) and intersecting a 
fixed straight line ; and suppose that points are taken on the rays, beyond the 
fixed straight line, such that the portions of the rays intercept^ between the 
fixed straight line and the point are equal to a constant distance (iidtmifia), 
the locus of the points is a conchoid which has the fixed straight line for 
asymptote. If the "distance " a is measured fi-om the intersection of the ray 
with the given straight line, not in the direction awa^ from the pole, but 
towards the pole, we obtain three other curves accordmg as a is less than, 
equal to, or greater than ^, the distance of the pole from the fixed straight line, 
which is an asymptote in each case. The case in which a>d gives a curve 
which forms a loop and then proceeds to infinity in the way Proclus describes. 
Now we know both from Eutocius (Comm. an Archimedes^ ed. Heiberg, iii. 
p. 114) and Proclus (p. 272, 3 — 7) that Nicomedes wrote on conchoidr (in 
the plural) and Pappus (iv. p. 244, 18) says that besides the "first" (used as 
above stated) there were "the second, the third and the fourth which are 
useful for other theorems.'' 

(2) Proclus next observes (p. 177, 9) that, of the lines which extend 
without limit, some are ^^ asymfioiic** (davfwrTWToi), namely "those which 
never meet, however they are produced," and some are ^^ symptotic^** namely 
"those which will meet sometime"; and, of the "asymptotic" class, some 
are in one plane, and others not Lastly, of the "asymptotic" lines in one 
plane, some preserve always the same distance firom one another, while others 
continually "lessen the distance, like the hyperbola with reference to the 
straight line, and the conchoid with reference to the straight line." 

^ Geminus' second classification. 

I This (from Proclus, pp. iii, 9 — 20 and 112, 16—18) can be shown in a 
diagram thus : 

Incomposite lines 
I dffit^BiTOi ypa/iiuU 

f I ' I 

' simple, irXfj mixed, fuicHj 

1 . I 



making a figure indeterminate 

(e.g. circle) (straight line) 



I ' 1 

lines in planes lines on solids 

I td h roit CT€ptott 

n 



line meeting itself extending without limit 

(e.g. cissoid) 



lines formed by sections lines round solids 

aX Karii rdt roftdt al w€pl rd rr€p€d 

(e.g. conic sections, spiric curves) (e.g. heiix about a sphere or about a cone) 

(cylindrical helix) (all others) 

Notes on classes of "lines" and on particular curves. 

We will now add the most interesting notes found in Proclus with 
reference to the above classifications or the particular curves mentioned. 



H. E. II 



L 



x6a BOOK I [i. DV. t 

1. Homoeomeric lines. 

By this tenn {ifunofuptU) are meant lines which are alike in all partii so 
that in any one such curve any part can be made to coincide with any odier 
part Proclus observes that these lines are only three in number, two being 
''simple'' and in a plane (the stra^ht line anid the circle^ and the dura 
''mixed," (subsisting) "alxMit a sohd,** namdy the cylindrical hdiz. The 
latter curve was also called the nxkiias or tufckUom^ and its k^moemmnc 
property was proved by ApoUonius in his woric wiyn tov K^xkiom (Prochis, 
p. 105, 5). The fact that mere are only three kamoeomiru lines was proved 
by Geminus, "who proved, as a preliminary proposition, that, if from a point 
{M Tov o^ficiov, but on p. 251, 4 iu^ Mt oiyfMiov) two stnuf^t lines be orawn 
to a homoeomeric line making equal angles with it, the straight Hnei are 
equal" (pp. 112, 1— 113, 3. cfc P- ^S't 2—19)- 

2. Mixed lines. 

It might be supposed, says Proclus (p. 105, xxX that the cylindrical hdiz, 
being homoeomeric^ like the straight liiMS and the cirde, must like them be 
simple. He replies that it is not smiple, but mixei^ because it is generated by 
two unlike motions. Two like motions, said Geminus, eg. two motions at the 
same speed in the directions of two adjoining sides of a sc^uare, produce a 
simple hne, namely a straight line (the diagonal); and again, if a straight line 
moves with its extremities upon the two sides of a right an^e respectively, i 
this same motion gives a simpk curve (a circle) for tte locus of the midme J 
point of the straight line, and a mUxii curve (an dlipae) for the locus of any I 
other point on it (p. 106, 3 — 15). i 

Geminus also explained that the term ^ mixed," as ai^lied to curves, and 
as applied to surfaces, respectively, is used in different senses. As apf^ed to | 
curves, "mixing" neither means simple "putting together ** ((nMco-it) nor 
" blending " (Kpcuric). Thus the helix (or spiral) is a " mixed " Une, but (i) it j 
is not " mixed '* in the sense of "putting together," as it would be if, say, part ] 
of it were straight and part circular, and (2) it is not mixed in the sense of 
" blending," beoiuse, if it is cut in any way, it does not present the appearance 
of any simple lines (of which it might be supposed to be compounded, as it \ 
were). The " mixing " in the case of lines is rather that in which the con- 
stituents are destroyed so far as their own character is concerned, and are 
replaced, as it were, by a chemical combination {yrrw iv avrj avpt^apfiiva rk 
oKpa KoX inrvK€xiffjJva). On the Other hand " mixed " surfaces are mixed in 
the sense of a sort of " blending " (icara rti^a Kpcuriv). For take a cone gene- ' 
rated by a straight line passing through a fixed point and passing always j 
through the circumference of a circle : if you cut this by a plane puallel to ^ 
that of the circle, you obtain a circular section, and if you cut it by a plane 
through the vertex, you obtain a triangle, the " mixed " surface of the cone | 
being thus cut into simple lines (pp. 117, 22 — 118, 23). 

3. Spiric curves. 

These curves, classed with conies as being sections of solids, were dis- 
covered by Perseus, according to an epigram of Eratosthenes quoted by 
Proclus (p. 112, i), which says that Perseus found "three lines upon (or, 
perhaps, m addition to) five sections" (rpcic ypa/ifia« M wivT€ to/muv). 
Proclus throws some light upon these in the following passages : 

"Of the spiric sections, one is interfaced, resembling the horse-fetter j 
{hntov wi&ti) ; another is widened out in the middle and contracts on each 



1! 



I.DEF. a] NOTE ON DEFINITION d 163 

side (of the middle), a third is elongated and is narrower in the middle, 
broadening out on each side of it" (p. 112, 4 — 8). 

"This is the case with the spiric sutface \ for it is conceived as generated 
by the revolution of a circle remaining at right angles [to a plane] and turning 
about a point which is not its centre [in other words, generated by the revo- 
lution of a circle about a straight line in its plane not passing through the 
centre]. Hence the spire takes three forms, for the centre [of rotation] is 
either on the circumference, or within it, or without it. And if the centre of 
rotation is on the circumference, we have the continuous spire (owfxi;?), if 
within, the interlaced (ifjLV9ir\€yfuvyj\ and if without, the open (Sicxi/f). And 
the spiric sections are three according to these three differences'' (p. 119, 
8-17). 

" When the Aippopede, which is one of the spiric curves, forms an angle 
with itself, this angle also is contained by mixed lines" (p. 127, i — 3). 

"Perseus showed for spirics what was their property ((rvfiirrcDfui) *' 
(P- 356, "). 

Thus the spiric surface was what we call a tore, or (when open) an anchor- 
ring. Heron (Def 98) says it was called alternatively spire (cnrcipa) or ring 
(fcpocos); he calls the variety in which "the circle cuts itself," not "interlaced," 
but "crossing-itself" (ffvaXXarrovo-a). 

Tannery^ has discussed these passages, as also did Schiaparelli*. It is clear 
that Proclus' remark that the difference in the three curves which he mentions 
corresponds to the difference between the three surfaces is a slip, due perhaps 
to too hurried transcribing from Geminus : all three arise from plane sections 
of the open anchor-ring. If r is the radius of the revolving circle, a the 
distance of its centre from the axis of rotation, d the distance of the plane 
section (supposed to be parallel to the axis) from the axis, the three curves 
described in the first extract correspond to the following cases : 

(i) d^a-r. In this case the curve is the hippopede^ of which the 
lemniscate of Bernoulli is a particular case, namely that in which a = 2r. 

The name hippopede was doubtless adopted for this one of Perseus' curves 
on the ground of its resemblance to the hippopede of Eudoxus, which seems to 
have been the curve of intersection of a sphere with a cylinder touching it 
internally. 

i2) fl + r > //><!. Here the curve is an ovaL 
3) a>d>a-r. The curve is now narrowest in the middle. 
Tannery explains the "three lines upon (in addition to) five sections" 
thus. He points out that with the open tore there are two other sections 
corresponding to 

(4) d= a : transition from (2) to (3). 

(5) fl-r>^>o, in which case the section consists of two symmetrical 
ovals. 

He then shows that the sections of the closed or continuous tore^ corre- 
sponding to a = r, give curves corresponding to (2), (3) and (4) only. Instead 
of (i) and (5) we have only a section consisting of two equal circles touching 
one another. 

On the other hand, the third spire (the interlaced variety) gives three new 
forms, which make a group of three in addition to the first group oifive sections. 

> Pour rhistoire da Hgtus et surfaces courbts dams rasttiquitl in BulUtin des sciences 
mathim, et astronom. vili. (1884), pp. 35 — 37. 

' Die kofmoceniriscken Spkarm des Eudoxus^ des Kallippus und des ArisioteUs (Ahhrnnd- 
lungm tstr Geuk, der Maik. I. Heft, 1877, pp. 149 — 151. 

II — 2 






L-— 



164 



BOOK I 



[l. DEP. 



The difficulty which I see in this interpretation is the fiict that, just after 
"three lines on five sections " are mentioned, Produs describes three curves 
which were evidently the roost important ; but these three belong to three of 
the five sections of the open tore^ and are not separate from them. 

4. The cissoid. 

This curve is assumed to be the same as that by means of which, accordirig 
to Eutodus {Comm. on Atxkimiies^ in. p. 79 sqq.^ Diodes in his book w^ 
irvfMiv {On iuming-gkuses) solved the problem of doubling the cube. It is 
the locus of points which he found b^ the following construction. Let AC^ 
BD be diameters at right angles in a arcle with centre A 

Let E^ J^be points on the quacbants BC^ BA respectivdy such that the 
arcs BE^ BE axe equal. 

Draw EG, Elf perpendicular to CA. 
Join AE^ and let F be its intersecticm 
with Eir. 

The dssoid is the locus of all the 
points E corresponding to different posi- 
tions of E on the quadrant BC and of E 
at an equal distance from B along the arc 
BA. 

^ is the pointy the curve correspond- 
ing to the position C for the point JE*, and 
B the point on the curve correqKmding 
to the position of E in which it ooinddes 
with A 

It is easy to see that the curve extends 
in the direction AB beyond B, and that 
CJ^ drawn perpendicular to CA is an 
asjrmptote. It may be regarded also as 
having a branch AD symmetrical with 
ABf and, beyond D^ approaching EC produced as asjrmptote. 

If OA, OD are coordinate axes, the equation of the curve is obviously 

where a is the radius of the drcle. 

There is a cusp at A, and it agrees with this that Proclus should say 
(p. 126, 24) that "dssoidal lines converging to one point like the leaves of 
ivy — for this is the origin of their name — form an angle." He makes the 
slight correction (p. 128, 5) that it is not two parts of a curve, but am curve, 
which in this case makes an angle. 

But what is surprising is that Produs seems to have no idea of the curve 
passing outside the drcle and having an asymptote, for he several times 
speaks of it as a closed curve (forming a figure and including an area): cf. 
p. 152, 7, "the plane (area) cut off by the cissoidal line has one bounding 
(line), but it has not in it a centre such that all (straight lines drawn to the 
curve) fix>m it are equal." It would appear as if Proclus regarded the cissoid 
as formed by ^efour symmetrical dssoidal arcs shown in the figure. 

Even more pecub'ar is Proclus' view of the 

5. *• Single-turn Spiral." 

This is really the spiral of Archimedes traced by a point starting fix>m 
the fixed extremity of a straight line and moving uniformly along it, while 




1 



I. DEFF. 2—4] NOTES ON DEFINITIONS 2—4 165 

simultaneously the straight line itself moves uniformly in a plane about its fixed 
extremity. In Archimedes the spiral has of course any number of turns, the 
straight line making the same number of complete revolutions. Yet Proclus, 
while giving the same account of the generation of the spiral (p. 180, 8 — 12), 
regards the single-turn spiral as actually stopping short at the point reached 
after one complete revolution of the straight line : " it is necessary to knaw 
that extending without limit is not a property of all lines; for it neither 
belongs to the circle nor to the cissoid, nor in general to lines which form 
figures ; nor even to those which do not form figures. For even the single- 
turn spiral does not extend without MmiX— far it is constructed between two 
points — nor does any 0/ the other lines so generated do so'* (p. 187, 19 — 25). 
It is curious that Pappus (viii. p. mo sqq.) uses the same term fiovoa-Tpo^ 
2Ai^ to denote one turn, not of the spiral, but of the cylindrical helix. 

Definition 3. 

Tpofjififj^ ik wipara irqiiMia, 

The extremities of a line are points. 

It being unscientific, as Aristotle said, to define a point as the '* extremity 
of a line " («pas ypafifi^^)^ thereby explaining the prior by the posterior, 
Euclid defined a point differently; then, as it was nec^^Bary to connect a 
point with a line, he introduced this escplanation after the definitions of both 
had been given. This compromise is no doubt his own idea; the same 
thing occurs with reference to a surface and a line as its extremity in Def. 6, 
and with reference to a solid and a surface as its extremity in xi. Def. 2. 

We miss a statement of the facts, equally requiring to be known, that a 

** division " ^Stcupco-ts) of a line, no less than its " begmning ** or *' end," is a 

point (this is brought out by Aristotle: cf. Jfetaph, 1060 b 15), and that 

the intersection of two lines is also a point If these additional explanations 

had been given, Proclus would have been spared the difiSculty which he finds 

in the fact that some of the lines used in Euclid (namely infinite straight lines 

on the one hand, and circles on the other) have no "extremities." So also 

the ellipse, which Proclus calls by the old name Bvptos (" shield "). In the 

case of the circle and ellipse we can, he observes (p. 103, 7), take a portion 

I bounded by points, and the definition applies to that portion. His rather 

I far-fetched distinction between two aspects of a circle or ellipse as a line and 

* as a closed figure (thus, while you are describing a circle, you have two extremi- 

< ties at any moment, but they disappear when it is finished) is an unnecessarily 

elaborate attempt to establish the literal universality of the "definition," 

which is really no more than an explanation that, if a line has extremities, 

those extremities are points. 

Definition 4. 

A straight line is a line which lies evenly with the points on itself 

The only definition of a straight line authenticated as pre-Euclidean is 
that of Plato, who defined it as " that of which the middle covers the ends " 
(relatively, that is, to an eye placed at either end and looking along the 
straight line). It appears in the Farmenides 137 s : "straight is whatever has 
its middle in fi'ont of (Le. so placed as to obstruct the view of) both its ends " 



x66 BOOK I [i.Dir.4 

(cMv y€ ov ivri fuo-ovififi^o&roir J^TOCV.Mvpotf^crf). Alistode ^poCtt it in 
equivalent terms (72^/W vi. ix, 148 b 2j\ oS ri jUnv iw u rfi 9 $ u row wipmn; 
and, as he does not mention the name of its author, but states it in combina- 
tion with the definition of a line as the extremity of a surbce, we vm assume 
that he used it as being weU known. Produs also quotes the definition as 
ipiato's in almost identictd termSi' i% ra gUm roUiKpoii9iw a r pa o$u(p. X091 ai). 
This definition is ingenious, but implicitly appeals to the sense of sight and 
involves the postulate that the line of sight is straiffht (Cf. the A ri stotel i an 
Problems 31, 20, 959 a 39, where the question is why we can better observe 
straightness in a row, say, of letters with one eye than with twa) As nqguds 
the straightness of ''visual iEys,'*a^cii^ d Euclid's own Q^tks, De£ x, a, 
assumed as hypotheses^ in which he first speaks of the ''stnuj^t lines" drawn 
from the eye, avoiding the word jfcis, and then sa^ that the figure contained 
by the visual rays (o^ces) is a cone with its vertex m the eye« 

As Aristotle mentions no definition of a straight line resembling EucUd's, 
but gives only Plato's definition and the other explaining it as the ''extremity 
of a surface,'' the latter being evidently the current defimtion in contemporary 
textbooks, we may safely infer that Euclid's definition was a new departure oif 
his own. 

Proclus on Euclid's definition. 

Coming now to the interpretation of Euclid's definition, cMm yni^ 
iariVf ipif i( wov roi« i^* javr^ mffuioft kuto^ we find any number of sUghtly 
different versions, but none that can be described as quite satisfiictory ; some 
authorities, e.g. Savile, have confessed that they could make nothing of it It 
is natural to appeal to Proclus first ; and we find that he does in net give an 
interpretation which at first sight seems plausible. He says (p. X09, 8 sq.) that 
Euchd "shows by means of this that the straight line alone [of all lines] 
occupies a distance (icarcxciv Su^on^/ia) equal to that between the points on it 
For, as fiar as one of the points is distant from another, so great is the length 
(f&^yc^os) of the straight line of which they are the extremities ; and this b the 
meaning of lying i( urov to (or with) the points on it " \i£ ttrw being thus, J 
apparently, interpreted as "at" (or "over") "an equal distance"]. "But if \ 
you take two points on the circumference (of a circle) or any other line, the f 
distance cut off between 'them along the line is greater than the interval 
separating them. And this is the case with every line except the straight line. 
Hence the ordinary remark, based on a common notion, that those who 
journey in a straight line only travel the necessary distance, while those who 
do not go straight travel more than the necessary distance." (Cf. Aristotle, j 
De catlo i. 4, ayi a 13, "we always .call 'the distance of anything the straight 1 
line " drawn to it) Thus Proclus would interpret somewhat in this way : "a J 
straight line is that which represents extension equal with (the distances 
separating) the points 'on it" This explanation seems to be an attempt to i 
graft on to Euclid's definition the assumption (it is a Aofi/Savd/icvor, not a 
definition) of Archimedes {On the sphere and cylinder i. ad init) that "of all < 
the lines which have the same extremities the straight line is least" For this , 
purpose li Zcrov has apparently to be tiCken as meaning "at an equal distance," 
and again "lying at an e<^ual distance" as equivalent to "extending over (or * 
representing) an equal distance." This is difficult enough in itself, but is 
seen to be an impossible interpretation when applied to the similar definition 
of a plane by Euclid (Def. 7) as a surfece "which lies evenly with the straight 
lines on itself." In that connexion Proclus tries to make the same words If Zirou 



t 



J 



' 



I. DEF. 4] NOTE ON DEFINITION 4 167 

Mirai mean "extends over an equal area with." He says namely (p. 117, 2) 
that ''if two straight lines are set out" on the plane, the plane surface 
'' occupies a space equal to that between the straight lines." But two straight 
lines do not determine by themselves any space at all ; it would be necessary 
to have a closed figure with its boundaries in the plane before we could arrive 
at the equivalent of the other assumption of Archimedes that ''of surfaces 
which have the same extremities, if those extremities are in a plane, the plane is 
the least [in areal." This seems to be an impossible sense for l^ txrov even on. 
the assumption tnat it means "at an equal distance" in the present definition. 
The necessity therefore of interpreting li Sorov similarly in both definitions 
makes it impossible to regard it as referring to distance or length at all. It 
should be added that Simplicius gave the same explanations as Proclus 
(an-Nairld, p. 5). 

' The language and construction of the definition. 

Let us now consider the actual wording and grammar of the phrase igfris ii 
WW TOic i^ Iavn7f mffjAioi^ Kcirai. As regards Sxe expression ii lo-ov we note 
that Plato and Aristotle (whose use of it seems typical^ commonly have it in 
the sense of "on a footing of equality": cf. 61 1( law m Plato's Laws 777 d, 
919 d; Aristotle, Politics 1259 b 5 l{ lo-ov ctvat Povkerai r^v ^vcriv, "tend to 
be on an equality in nature," £tA, Nic. viii. 12, 1 161 a 8 IvravOa a-ovrcc i^ 
Iffw^ " there all are on a footing of equality." Slightly different are the uses 
in Aristotle, Eth. Nic. X. 8, 11 78 a 25 rcSv /icv y^ avayKotW xp«ta ical l{ Ztrov 
loTo^ "both need the necessaries of life to the same extent^ let us say"; Topics ix. 
15, I74a32^ lirov woiovvra rrfv ipiDTtfiny, "asking the question indifferently" 
(i.e. without showing any expectation of one answer being given rather than 
another). The natural meaning would therefore appear to be "evenly placed" 
(or balanced), "in equal measure," "indifferently" or "without bias" one way 
or the other. Next, is the dative roU ^^* iavrq^ (n;/icioi$ constructed with i( urov 
or with Kfirai? In the first case the phrase must mean "that which lies evenly 
with (or in respect to) the points on it," in the second apparently "that which, 
in (or by) the points on it, lies (or is placed) evenly (or uniformly)." Max Simon 
takes the first construction to give the sense "die Gerade liegt in gleicher 
Weise wie ihre Punkte." If the last words mean " in the same way as (or in 
like manner as) its points," I cannot see that they tell us anything, although 
Simon attaches to the words the notion o( distance (Abstand) like Proclus. 
The second construction he takes as giving "die Gerade liegt fiir (durch) ihre 
Punkte gleichmassig," "the straight line lies symmetrically for (or through) its 
points"; or, if «c€irai is taken as the passive ofri^/u, "die Gerade ist durch 
ihre Punkte gleichmassig gegeben worden," "the straight line is symmetrically 
determined by its points." He adds that the idea is here direction, and that 
both direction and distance (as between two different given points simply) 
would be to Euclid, as later to Bolzano (Betrachtungen Oder einige Gegenstdnde 
der Elementargeometriey 1804, quoted by Schotten, Inhalt und Methode des 
planimetrischen Unterrichts, 11. p. 16), primary irreducible notions. 

While the language is thus seen to be hopelessly obscure, we can safely 
say that the sort of idea which Euclid wished to express was that of a line 
which presents the same shape at and relatively to all points on it, without 
any irr^ular or unsymmetrical feature distinguishing one part or side of it 
from another. Any such irr^ularity could, as Saccheri points out (Engel and 
Stackel, Die Theorie der ParalleUinien von Euklid bis Gauss, 1895^ p. 109X be 
at once made perceptible by keeping the ends fixed and turning the line about 



i68 BOOK I [ldbf. 

them right round; if any two positions were distinguishable, e.g. one being to 
the left or right relatively to another, "it would not lie in a uniform manner 
between its points." 

A conjecture as to its origin and meaning. 

The question arises, what was the origin of Euclid's definition, or, how 
was it suggested to him ? It seems to me that the basis of it was really 
Plato's definition of a straight line as "that line the middle of which covers 
the ends." Euclid was a Platonist, and what more natural than that he 
should have adopted Plato's definition in substance, while regarding it as 
essential to change the form of words in order to make it independent of any 
implied appeal to vision, which, as a physical fiict, could not properly find a 
place in a purely geometrical definition? I believe therefore that Eudid's 
definition is simply an attempt ^albeit unsuccessful, fix>m the nature of the 
case) to express, in terms to which a geometer could not object as not being 
part of geometrical subject-matter, the same thing as the Platonic definition. 

The truth is that Euclid was attempting the impossible. As Pfleiderer 
says (Scholia to Euclid), " It seems as though the notion of a straight Ime^ 
owing to its simplicity, cannot be explained by any regular definition which 
does not introduce words already containing m themsislv^ by implication, 
the notion to be defined (such eg. are duection, equali^, uniformity or 
evenness of position, unswerving course), and as though it were impossible^ if 
a person does not already know what the term siraigki here means, to teadi 
it to him unless by putting before him in some way a pictuie or a dtrawing of 
it" This is accordingly done in such books as Veronese's Eiemtnii H 
geofnetria (Part i., 1904, p. 10): ''A stretched string,* e.g. a plumniet, a ray of 
light entering by a small hole into a dark room, are rutiUmal objects. The 
image of them gives us the abstract idea of the limited line which is called a 
rectilineal segment!^ 

Other definitions. 

We will conclude this note with some other fan^ous definitions of a straight 
line. The following are given by Proclus (p. no, 18—23). 

I. A fine stretched to the utmost^ itr oKpov rcro/icn; ypa/A/Aif. This appears 
in Heron (ist c a.d.) also, with the words "towards the ends" (hri ra a^^ra) 
added. (Heron, ed. Hultsch, Def. 5, p. 8). 

a. Part of it cannot he in the assumed plane while part is in one higher up 
i^ fur€iap€T4pff). This is sl proposition in Euclid (xi. i). 

3. jiU its parts Jit on all (other parts) alike, wavra avrfs rot fUfni www 
ofiolia^ iffMpfwiti, Heron has this too (Def. 5), but instead of "alike" he 
says iravTotm, "in all ways," which is better as indicating that the applied part 
may be applied one way or the reiferse way, with the same result 

4. That line which^ when its ends remain fixed^ itself remains fixed^ ri rw 
w€paTW¥ fLcvoWttiv icttt aMi fiivovau. Heron's addition to this, " when it is, as 
it were^ turned round in the same plane*^ (olov h rf avnp lwvw&^ irrp€^ofUyti\ 
and his next variation, "and about the same ends having always the same 
position," show that the definition of a straight line as "that which does 
not change its position when it is turned about its extremities (or any two 
points in it) as poles" was no original discovery of Leibniz, or Saccheri, or 
Krafll, or Gauss, but goes back at least to the banning of the Christian era. 
Gauss' form of this definition was: "The line in which lie all points that, 
during the revolution of a body (a part of space) about two fixed points, 
maintain their position unchanged is called a straight line." SchoUen 



7\ 



1. DEFr. 4, S] NOTES ON DEFINITIONS 4. S 169 

(i* P- 315) maintains that the notion of a straight line and its property of 
being determined by two points are unconsciously assumed in this definition, 
which is therefore a logical ''circle/' 

5. Tkat line which with one other of the same species cannot complete a 
figure^ 17 /xrra r^s ^fiociSovs fuas o-x/ffAa firi dmrtkowra. This is an obvious 
SaT€poy'irp6T€poy, since it assumes the notion of a figure. 

Lastly Leibniz' definition should be mentioned: A straight line is one 
which divides a plane into two halves identical in all but position. Apart from 
the fact that this definition introduces the plane, it does not seem to have any 
advantages over the definition last but one referred to. 

Legendre uses the Archimedean property of a straight line as the shortest 
distance between two points. Van Swinden observes {Elemente der Geometries 
1^341 P- 4)1 that to take this as the definition involves axjf#m//7^ the'proposition 
that any two sides of a triangle are greater than the third and proving that 
straight lines which have two points in common coincide throughout their 
length (cf- Legendre, Aliments de Geometric 1. 3; 8). 

The above definitions all illustrate the observation of Unger {Die Geometric 
des Euklidy 1833) : ^^ Straight is a simple notion, and hence all definitions of 
it must fail.... But if the proper idea of a straight line has once been grasped, 
it will b^ recognised in all the various definitions usually given of it ; all 
the definitions must therefore be regarded as explanations^ and among them 
that one is the best from which further inferences can immediately be drawn 
as to the essence of the straight line." 

Pefinition 5. 

A surface is that which has length and breadth only. 

The word ^irt^vcui was used by Euclid and later writers to denote surface 
in general, while they appropriated the word cViirffSov for plane surface, thus 
making htlw&w a species of the gentis iwif^tia, A sob'ts^ use of cn-i^oFcia 
by Euclid when a plane is meant (xi. Def. 1 1) is probably due to the fact that 
the particular definition came from an earlier textbook. Proclus (p. 116, 17) 
remarks that the older philosophers, including Plato and Aristotle, used the 
words lwi^v€ia and criircSov indifferently for any kind of surface. Aristotle 
does indeed use both words for a surface, with perhaps a tendency to use 
ciri^aycia more than criVcSov for a surface not plane. Cf. Categories 6, 5 a i sq., 
where both words are used in one sentence: "You can find a common 
boundary at wiiich the parts fit together»,a point in the case of a line, and a line 
in the case of a surface (cin^Vcia); for the parts of the surface (cViircSov) do fit 
together at some common boundary. Similarly also in the case of a body you 
can find a common boundary, a line or a surface (ciri^oi^cia), at which the 
parts of the body fit together." . Plato however does not use ciri^Vcca at all in 
the sense of surface, but only cv-iircSov for both surface and plane surfau. 
There is reason therefore for doubting the correctness of the notice in 
Diogenes Laertius, in. 24, that Plato "was the first philosopher to name, 
among extremities, ^'t plane surface " (cViircSos cv-c^ovcta). 

ffVi^cca of course means literally the feature of a body which is apparent 
to the eye (cirt^nfc), namely the surface. 

Aristotle tells us {De sensu 3, 439 a 31) that the Pythagoreans called a 
surface XP®^ which seems to have meant skin as well as colour, Aristotle 
explains the term with reference to colour (xp«i)f^) as a thing inseparable from 
the extremity (ir^«) of a body. 




X70 BOOK I 

Alternative definitions. 

The de6nitions of a surfiu» correspond to diose of a line. As in Aristotle 
a line is a magnitude ''(extended) one way, or in one 'dimension'** (c^* V^ 
"continuous one way" (c^' tr av¥€xk\ or *' divisible in one way" OMpaxJ 
Scoipcroi'), so a surface is a masnitude extended or continuous iwa ways (m 
Svo), or divisible in two ways (oixp). As in Euclid a surftce has "length and 
breadth '' only, so in Aristotle " breadth " is characteristic of the surface and is 
once used as synonymous with it (Meiafh. X020 a laX and again ''lengths 
are made up of long and short, surfaas of broad amd narrow^ and solids (<7km) 
of deep and shallow " (Meiaph. 1085 a 10). 

Anstotle mentions the common remark that a Une by Us moHon produces a 
surface {De anima i. 4, 409 a 4). He alsogives the a posteriori deKripti<Hi of 
a surface as the "extremitv of a solid** (Topics vi. 4, 141 b aa), and as "the 
section (rofiif) or division (fiuJpmn^) of a bo^" (ifetaph* 1060 b 14). 

Proclus remarks (p. 114^ ao) that we get a notion of a surfiure when we 
measure areas and mark their boundaries in the sense of length and breadth ; 
and we further get a sort of perception of it by looking at shadows, since 
these have no depth (for they do not penetrate the earth) but only have leqg^ 
and breadth. * - 

Classification of surfaces. 

Heron gives (Def. 75, p. 23, ed. Hultsch) two alternative divisions of 
surfaces into two classes, corresponding to Geminus* alternative divisions of 
lines, viz. into (i) incomposite and composite and (2) single and mixed. 

(i) Incomposite surfaces are ^ those which, when produced, fidl into (or 
coalesce with) themselves" (&rac UfioXkiiLwu a^roi tnJf iavni^ wArroiwtr), 
i.e. are of continuous curvature, e.g. the sphere. 

Composite surfaces are "those which, when produced, . cut one another." 
Of composite surfaces, again, some are (a) made up of non-homogeneous 
(elements) (c^ &,vo/ioiay€y£v) such as cones, cylinders and hemispheres, others 
lb) made up of homogeneous (elements), namely the rectilineal (or polyhedral) 
surfaces. 

(2^ Under the alternative division, simp/e surfaces are the plane and the 
sphencal surfaces, but no others ; the mixed class includes all other sur&ces 
whatever and is therefore infinite in variety. 

Heron specially mentions as belonging to the mixed class (a) the surfoce 
of cones, cylinders and the like, which are a mixture of plane and circular 
(fjLucraX ff^ cirtirffSov koI vcpt^cpcm) and (b) spiric surfaces, which are "a mixture 
of two circumferences " (by which he must mean a mixture of two drcular 
elements, namely the generating circle and its circular motion about an axis in . 
the same plane). 

Proclus adds the remark that, curiously enough, mixed surfaces may arise* 
by the revolution either of simple curves, e.^. in the case of the spire^ or of 
mixed curves, e.g. the "right-angled conoid" from a parabola, "another 
conoid" from the hyperbola, the "oblong" (c«>cffiyiccv, in Archimedes vapa- 
fioiccs) and " flat " (ImrAarv) spheroids from an ellipse according as it revolves . 
about the major or minor axis respectively (pp. 119, 6 — 120, 2). The hamoeo- 
meric surfaces, namely those any part of which wiU coincide with any other 
part, are two only (the plane and the spherical surface), not three as in the case 
of lines (p. 120, 7). 



J 



1. DBFF. 6, 7] NOTES ON DEFINITIONS 5—7 171 

Definition 6. 

*Eiri^aFffia« Sc wipara ypCLfjifAoL 

The extremities of a surface are lines. 

It being unscientific, as Aristotle says, to define a line as the extremity of 
a surface, Euclid avoids the error of defining the prior by means of the 
posterior in this way, and gives a different definition not open to this 
objection. Then, by way of compromise, and in order to show the connexion 
between a line and a surface, he adds the equivalent of the definition of a line 
previously current as an explanation. 

As in the corresponding Def. 3 above, he omits to add what is made 
dear by Aristotle {Metaph. 1060 b 15) that a "division" (Staipco-is) or 
"section" (ro/ii;) of a solid or body is also a surface, or that the common 
boundary at which two parts of. a solid fit together {Categories 6, 5 a 2) 
may be a surface. 

Proclus discusses how the fact stated in Def. 6 can be said to be true of 
surfaces like that of the sphere "which is bounded (ircirffpa<rraO, it is true, but 
not by lines." His explanation (p. 116, 8 — 14) is that "if we take the surface 
(of a sphere), so far as it is extended two ways (8ixig Scooran;), we shall find 
that it IS bounded by lines as to length and breadth ; and if we consider the 
spherical surface as possessing a form of its own and invested with a fresh 
quality, we must regard it as having fitted end on to beginning and made 
the two ends (or extremities) one, t^ing thus one potentially only, and not in 
actuality." ^ 

Definition 7. ^.^JVoW-^-- '^' ^ *" 

*EiriircSoc hrij^mftijL iarw, igfrif i( laov reus ^^* iavrrji cv^ciacf Kcirat. 
A plane surfieice is a surface which lies evenly with the straight lines on 
itself 

The Greek follows exactly the definition of a straight line mutatis mutandis^ 
i.e.' with raif...cvdffiaif for Tor«...cn7/xciOi«. Proclus remarks that, in general, 
all the definitions of a straight fa'ne can be adapted to the plane surface by 
merely changing the ^nus. Thus, for instance, a plane surface is " a surface 
the middle of which covers the ends " (this being the adaptation of Plato's 
definition of a straight line). Whether Plato actually gave this as the defini- 
tion of a plane surface or not, I believe that Euclid's definition of a plane 
surface as lying evenly with the straight litus on itself was intended simply to 
express the same idea without any implied appeal to vision (just as in the 
corresponding case of the definition of a straight line). 

I As already noted under Def. 4, Proclus tries to read into Euclid's defini- 
tion the Archimedeao assumption that "of surfaces which have the same 
extremities, if those extremities are in a plane, the plane is the least" But, 

' as I have stated, his interpretation of the words seems impossible, although it 

I is adopted by Simplidus also (see an-Nairizi). 

I Anjcient alternatives. 

The other ancient definitions recorded are as follows. 
I I. The surface which is stretched to the utmost {iw oKpov rcro/icn;) : a 
1 definition which Proclus describes as equivalent to Euclid's definition ([on 
j Proclus' own view of that definition). Cf. Heron, Def. 11, "(a surface) which 

is right (and) stretched out " (opOi^ cSa-a dwoT€TafUinj\ words which he adds to 

Eudbid's definition. 



1 73 BOOK I [l DBF. 7 

2. 71k€ least surface among ail thast which have the same exiremiHet. 
Proclus is here (p. 1 17, 9) obviously quoting the Axchiaiedean assumfUom. 

3. A surface all the parts of which have the property ofJUtmg am {each 
other) (Heron, Def. 11). 

4. A surface such thai a straight Ume JUs on ail parts of it (Proclosi 
p. 117, 8), or such that the UraighiHne fits on iialiways^ i.e. however pboed 
(Proclus, p. 117, 20). 

With diis should be compared : 

5. "(^ plane surface is) such thai^ if a straight line peus through two 
points on it, the line coincides wholly with it at every spot^ all ways/* ie. however 
placed (one way or the reverse, no matter how), 1^ ^rctJoy tvo ai^iMim Sifnim 
cvtfffui, jcol oktf avn; icara wearra rovor wamitK i^apfiiCenu, (Heron, Defl il). 
This appears, with the words mtra wivra riww wrolm omitted, in Theon of 
Smyrna (p. 112, 5, ed. HillerX so that it goes back at least as fiur as the 
ist c A.D. It is of course the same as the dfefinition qommonly attributed to 
Robert Simson, and very widely adopted as a substitute for Euclid's. 

This same definition appears also in an-Nairizi (ed. Curtse, p. zo) who, 
after quoting Simplicius' explanation (on the same lines as Proclus') of the 
meaning of Euclid's definition, goes on to say that ''others defined the plane 
surface as that in which it is possible to draw a straight line from any pdnt ! 
to any other." ♦ 

DiflBculties in ordinary definitions. 

Gauss observed in a letter to Bessel that the definition of a plane surfiioe 
as a surface such that^ if any two points in it ie tahen^ the straight line joining 
them lies wholly in the surface (which, for short, we will call ''Simson*^" 
definition) contains more than is necessary, in that a plane can be obtained by 
simply projecting a straight line lying in it from a point outside the line but also 
lying on the plane ; in fact the definition includes a theorem, or postulate, as 
well. The same is true of Euclid's definition of a plane as the surface which 
" lies evenly with (all) the straight lines on itself^" because it is suffident for a 
definition of a plane if the surface *' lies evenly " with those lines onl^ which 
pass through a fixed point on it and each of the several points of a straight line 
also lying m it but not passing through the point But from Euclid's point 
of view it is immaterial whether a definition contains more than the necessary 
minimum provided that the existence of a thing possessing all the attributes 
contained in the definition is afterwards proved. This however is not done 
in regard to the plane. No proposition about the nature of a plane as such 
appears before Book xi., .although its existence is presupposed in all the 
geometrical Books i. — iv. and vi. ; nor in Book xi. is there any attempt to 
prove, e.g. by construction, the existence of a surface conforming to the 
definition. The explanation may be that the existence of the plane as defined 
was deliberately assumed from the beginning like that of points and lines, the 
existence of which, according to Anstotle, must be assumed as principles 
unproved, while the existence of everything else must be proved ; and it may 
well«be that Aristotle would have included plane surfaces with points and 
lines in this statement had it not been that he generally took his illustrations 
horn plane geometry (excluding solid). 

But, whatever definition of a plane is taken, the evolution of its essential 
properties is extraordinarily difficult. Crelle, who wrote an elaborate article 
Zur Theorie der Ebene (read in the Academic der Wissenschaften in 1834) of 
which account must be taken in any full history of the subject, observes that, 




I. DEF. 7] NOTE ON DEFINITION 7 173 

since the plane is the field, as it were, of almost all the rest of geometry, while 
a proper conception of it is necessary to enable Eucl. i. i to be understood, 
it might have been expected that the theory of the plane would have been the 
subject of at least the same amount of attention as, say, that of parallels. This 
however was far from being the case, perhaps because the subject of parallels 
(which, for the rest, presuppose the notion of a plane) is much easier than that 
of the plane. The nature of the difficulties as regards the plane have also 
been pointed out recently by Mr Frankland {The First Booh of Euclid's 
Elements^ Cambridge, 1905): it would appear that, whatever definition is 
taken, whether the simplest (as containing the minimum necessary to deter- 
mine a plane) or the more complex, e.g. Simson's, some postulate has to be 
assumed in addition before the fundamental properties, or the truth of the 
other definitions, can be established. Crelle notes the same thing as r^ards 
Simson's definition, containing more than is necessary. Suppose a plane in 
which lies the triangle ABC. Let AD join the vertex A 

^ to any point D on BC^ and BE the vertex B to any 

J point E on CA. Then, according to the definition, AD 
lies wholly in the plane of the triangle; so does BE. 
But, if both AD and BE are to lie wholly in the one 
plane, AD^ BE must intersect, say at F\ if they did not, 
there would be two planes in question, not one. But the fact that the lines 
intersect and that, say, AD does not pass above or below BE^ is by no 
means self-evident. 

Mr Frankland points out the similar difficulty as regards the simpler 
definition of a plane as the surface generated by a straight 
line passing always through a fixed point and always 
intersecting a fixed straight line. Let OPF^ OQQ 
drawn from O intersect the straight line X 2X P^ Q 
respectively. Let R be any third point on X\ then it 
^ needs to be proved that OR intersects P'Q in some ^ ^\ 

point, say R*. Without some postulate, however, it is 
not easy to see how to prove this, or even to prove that P'Q intersects X. 

Crelle's essay. Definitions by Fourier, Deahna, Becker. 

Crelle takes as the standard of a good definition that it shall be, not only as 
simple as possible, but also the best adapted for deducing, with the aid of the 
simplest possible principles, further properties belonging to the thing defined. 
He was much attracted by a very lucid definition, due, he says, to Fourier, 
according to which a platu is formed by the aggregate of ail the straight lines 
which^ passing through one point on a straight line in space^ are perpendicular 
to that straight line. (This is really no more than an adaptation from Euclid's 
proposition xi. 5, to the efiect tha^ if one of four concurrent straight lines be 
at right angles to each of the other three, those three are in one plane, which 
proposition is also used in Aristotle, Meteorologica in. 3, 373 a 13.) But 
Crelle confesses that he had not been able to deduce the necessary properties 
from this and had had to substitute the definition, already mentioned, of a 
plane as the surface containing^ throughout their whole lengthy all the straight 
lines passing through a fioud point and also intersecting a straight line in space \ 
and he only claims to have proved, after a long series of propositions, that the 
"Fourier"- or " perpendicular "-surface and the plane of the other definition 
just given ate identical, after which the properties of the " Fourier "-surface 
can be used along with those of the plane. The advantage of the Fourier 
definition is that it leads easily, by means of the two propositions that 







174 BOOK I [l 

triangles are equal in all respects (i) when two sides and the induded an^ 
are respectively equal and (2) when all three sides are respectively equal, to the 
property expressed in Siroson's definition. But Crelle uses to establish these 
two congruence-theorems a number of propositions about equal angles^ si^k- 
meniary angles, right angles, greater and Use angles ; and it is difficult to 
question the soundness of Schotten's criticisin that these notions in themselves 
really presuppose that of a plane. The difficulty due to Fourier's use of 
the word '' perpendicular,*' if that were all, could no doubt be got over. Thus 
Deabna in a dissertation (Marburg, 1837) constructed a plane as follows. 
Presupposing the notions of a straight line and a qphere, he observes that, if a 
sphere revolve about a diameter, all the points of its surftce which move 
describe closed curves (circles). Each of mese circles, during the revolution, 
moves along itself, and one of them divides the sur&ce of the qpheie into two 
congruent parts. The aggrc^te then of the lines joining the centre to the 
points of this circle forms ih^ plane. Again, J. K. Becker (Die ElemetUe der 
Geometries 1877) pointed out that the revolution of a right ang^e about one 
side of it produces a conical suifice which differs from all other conical 
surfaces generated by the revolution of oAer angles in the foct that the 
particular cone coincides with the cone vertically opposite to ii : this characteristic 
might therefore be taken in order to get rid of the use of the rig^ angfe. 

W. Bolyai and Lobachewsky. 

Very similar to Deahna's equivalent for Fourier's definition is the device 
of W. Bolyai and LobachewBky (described by Frischauf, Elemente ier 
absoluten Geometries 1876). They worked upon a fundamental idea first 
suggested, apparently, by Leibniz. Briefly stated, their wi^ of evolving a 
plane and a straight line was as follows. Conceive an infinite number of 
pairs of concentric spheres described about two fixed points in space, (7, <7, 
as centres, and with equal radii, gradually increasing : these pairs of equal 
spherical surfaces intersect respectively in homogeneous curves (circles), and > 
the '* Inbegriff *' or aggr^ate of these curves of intersection forms a plane. 
If ^ be a point on one of these circles {k say), suppose points 3f, M' to start 
simultaneously from A and to move in opposite directions at the same speed 
till they meet at BsSBLy, B then is *' opposite" to A^ and A^ B divide the 
circumference into two equal halves. If the points ^, ^ be held fast and the 
whole system be turned about them until O takes the place of 0^ and O of 
Os the circle k will occupy the same position as before (though turned a 
different way). Two opposite points, P^ Q say, of each of the other circles 
will remain stationary during the motion as well as A, B: the " Inbegriff " or 
aggregate of all such points which remain stationary forms a straight line. It 
is next observed that the plane as defined can be generated by the revolution 
of the straight line about Off^ and this suggests the following construction 
for a plane. Let a circle as one of the curves of intersection of the pairs of 
spherical surfaces be divided as before into two equal halves at A^ B. Let the 
arc ADB be similarly bisected at Z>, and let C be the 
middle point of AB. This determines a straight line CD 
which is then defined as ''perpendicular" to AB. The revo- 
lution of CD about AB generates a plane. The property 
stated in Simson's definition is then proved by means of the 
congruence-theorems proved in Eucl. i. 8 and i. 4. The 
first is taken as proved, practically by considerations of 
symmetry and homogeneity. If two spherical surfaces, not necessarily equal, 
with centres O^ O intersHSCt, A and its ''opposite" point B are taken as 




\ L DEF. 7] 



NOTE ON DEFINITION 7 



175 




before on the curve of intersection (a circle) and, relatively to Off^ the point 
A is taken to be convertible with B or any other point on the homogeneous 
curve. The second (that of Eucl. i. 4) is established by simple application. 
Rausenberger objects to these proofs on the 'grounds that the first assumes 
that the two spherical surfaces intersect in one single curve, not in several, 
and that the second compares ang/es : a comparison which, he says, is possible 
only in a p/ant^ so that a plane is really presupposed. Perhaps as regards 
the particular comparison of angles Rausenberger is hypercritical; but it is 
difficult to r^ard the supposed proof of the theorem of Eucl. i. 8 as sufficiently 
rigorous (quite apart from the use of the uniform mo/ion of points for the 
purpose of bisecting lines). 

Simson's property is proved from the two congruence-theorems thus. 
Suppose that AB is *' perpendicular" (as defined by Bolyai) to two generators 
CJff CN of a plane, or suppose CM^ CN respectively to make with AB two 
angles congruent with one another. It is enough to prove that, if Z' be any 
pomt on the straight line MN^ then CP^ just as 
much as CM^ CW respectively, makes with AB two 
angles congruent with one another and is therefore 
a generator. We prove successively the congruence 
of the following pairs of triangles : 

ACM, BCM 

ACN, BCN 

AMN, BMN 

AMP, BMP 

ACP, BCP, 
whence the angles ACP, BCPzx^ congruent 
Other views. 

Enriques and Amaldi (EUmenti di geometria, Bologna, 1905), Veronese 
(in his EUmenti) and Hilbert all assume as a postulate the property stated in 
Simson's definition. But G. Ingrami {Elementi di geometria, Bologna, 1904) 
proves it in the course of a remarkable series of closely argued proposition 
oased upon a much less comprehensive postulate. He evolves the theory of 
the plane from that of a triangle, beginning with a triangle as a mere three-side 
(trilatero), i.e. a frame, as it were. His postulate relates to the three-side and 
is to the effect that each " (rectilineal) segment " joining a vertex to a point of 
the opposite side meets every segment similarly joining each of the other two 
vertices to the points of the sides opposite to them respectively, and, con- 
versely, if a point be taken on a segment joining a vertex to a point of the 
opposite side, and if a straight line be drawn from another vertex to the point 
on the segment so taken, it will if produced meet the opposite side. A 
triangle is then defined as the figure formed by the aggregate of all the 
segments joining the respective vertices of a thru-side to points on the 
opposite sides. After a series of propositions, Ingrami evolves a plane as the 
figure farmed by the " half straight-lines " which project from an internal point 
^ the trian^e the points of the perimeter, and then, after two more theorems, 
proves that a plane is determined by any three of its points which are not in 
a straight line, and that a straight line which has tu^o points in a plane has all 
Us points in it. 

The argument by which Bolyai and Lobachewsky evolved the plane is 
of course equivalent to the definition of a plane as the locus ef all points 
eqmdistant from two fixed points in space. 



176 BOOK I [l mrr. 7—9 

Leibniz in a letter to Giordano defined a plane as ikai tmtfaa wUdk 
divides space into two congruemi parts. Admting to Giordano's cnticisni that 
you could conceive of surfaces and lines which divided qwce or a plane into 
two congruent parts without beipg/£iff^ or i/rw^A/ respectively. Bees (jOiar 
EuJUidische und NichtEuklidiseke Geameirie^ 1888) pointed out that what was 
wanted to complete the definition was the further condition that the two 
congruent spaces could be sHi tUong each other without the surfiuxs ceasiiy 
to coincide, and claimed priority for his completion of the definition in dus 
way. But the idea of all the parts of a i^ane fitting enctly on eUl other parts 
is ancient, appearing, as we have seen, in Heron, Det ii. 

Definitions 8, 9. 

8. 'EvfircSoc Sc ywFui coTcv 1} i» hnwAf tuo jpa§t§aAy iano^ubm^ AXXajXmw 
fcoi fi^ iw* fMciac K€ifjL€Viiiiy wpit oAX^Xac tmk ypafi§im¥ kkun^ 

9. ^Orav Sc al xi/Mcxovcnu r^ ytirior yfM^ngud Mumi Sau^, Mvypapt^un 
jcoXciTcu 17 yuyta. 

8. A plane angle is the inelisuttion to one another of two Hnes m a pkme 
which meet one another and do mot lie in a straight Usie. 

9. And when the lines containing the angit are straight^ the angle is tailed 
rectilineal. 

The phrase "not in a straight line** is strange, seeing that the definition 
purports to apply to angles formed by curves as well as straight lines. We 
should rather have expected eontmnous (om^xf^) with one another; and 
Heron takes this to be the meaning, since he at once adds an expknation as 
to what is meant by lines not being continuous (oA ovi^xm). It looks as though 
Euclid really intended to define a rectilineal angle, but on second thoughts^ 
as a concession to the then common recognition of curvilineal angles, altered 
^ straight lines " into " lines " and separated the definition into two. 

I think ril our evidence suggests that Euclid's definition of an angle as 
inclination (xXuric) was a new departure. The word does not occur in 
Aristotle ; and we should gather fi-om him that the idea generallv associated 
with an angle in his time was rather deflection or breahing of lines (xXiurtv) : cf. 
his common use of #cc#cAairtfai and other parts of the verb fcXoK, and also his 
reference to one bent line forming an angle (r^r KtKOfjL^iiyqy koX ^^oiNroy yvn^iav^ 
Jfetaph, 1016 a 13). 

Proclus has a long and elaborate note on this definition, much of which 
(pp. 121, 12 — 126, 6) is apparently taken direct fi-om a work by his master 
Syrianus (6 iJ/«^cpos Ka$frf€fjMv). Two criticisms contained in the note need 
occasion no difficulty. One of these asks how, if an angle be an inclination, 
one inclination can produce two angles. The other (p. 128, 2) is to the effect 
that the definition seems to exclude an angle formed by one and the same 
curve with itself, e.g. the complete a'ssoid [at what we call the '^ cusp "1 or the 
curve known as the hippopede (horse-fetter) [shaped like a lemniscatej. But i 
such an "angle" as this belongs to higher geometry, which Euclid may well 
be excused for leaving out of account in any case. 

Other ancient definitions : ApoUonius, Plutarch, Carpus. 

Proclus* note records other definitions of great interest ApoUonius 
defined an angle as a contracting of a surface or a solid at one Ooint under a 
hrohen line or surface (onnayory^ crt^oyciav ^ oripcov 9po% l¥i oiffMcIy vwh 
K€KXaafUtqg ypo/^v ^ ^t^omigi), where again an angle is supposed to be 
formed by one broken line or surface. Still more interesting, perhaps, is the 
definition by "those who say that the first distance under the point (to vywrov 



I 



I. Dkff. 8, 9] NOTES ON DEFINITIONS 7—9 177 

iiaarrifia vtrb ro ainUiov) is the angle. Among these is Plutarch, who insists 
that Apollonius meant the same thing ; for, he says, there must be some first 
distance under the breaking (or deflection^ of the including lines or surfaces, 
though, the distance under the point bemg continuous, it is impossible to 
obtain the actual firsty since every distance is divisible without limit " (cir* 
axc(f>ov). There is some vagueness in the use of the word " distance" {^^Mjtm\^ \ 
thus it was objected that ** if we anyhow separate off the/rj/" (distance being 
apparently the word understood) ''and draw a straight Ime through it^ we get 
a triangle and not one angle.'* In spite of the objection, I cannot but see in 
the idea of Plutarch and the others the germ of a valuable conception in 
infinitesimals, an attempt (though partial and imperfect) to get at the rate 
of divergence between the lines at their point of meeting as a measure of the 
angle between them. 

A third view of an angle was that of Carpus of Andoch, who said " that 
the angle was a quantity (itoctof), namely a distana {Stdarrifia) between the 
lines or surfaces containing it. This means that it would be a distance (or 
divergence) in one sense (c^* ty Sccorctfc), although the angle is not on that 
account a straight line. For it is not everything extended in one sense (to ^^' tv 
ScaoraroV) that is a line." This very phrase " extended one way" being held 
to define a line^ it is natural that Carpus' idea should have been described as 
the greatest possible paradox (Trarroii^ irapaSo^orarov). The difficulty seems to 
have been caused by the want of a different technical term to express a new 
idea ; for Carpus seems undoubtedly to have been anticipating the more 
modem idea of an angle as representing divergence rather than distance, and to 
have meant by c^' tv in one sense (rotationally) as distinct from one way or in 
one dimension (linearly). 

To what category does an angle belong? 

There was much debate among philosophers as to the particular category 
(according to the Aristotelian scheme) in which an angle should be placed ; 
is it, namely, a quantum (^fifT6v\ quale (iroiov) or relation (n-pos n) ? 

1. Those who put it in the category o{ quantity argued from the fact that 
a plane angle is divided by a line and a solid angle by a surface. Since, then, 
it is a surface which is divided by a line, and a solid which is divided by 
a surface, they felt obliged to conclude that an angle is a surface or a solid, and 
therefore a magnitude. But homogeneous finite magnitudes, e.g. plane 
angles, must bear a ratio to one another, or one must be capable of being 
multiplied until it exceeds the other. This is, however, not the case with a 
rectilineal angle and the horn-like angle (iccparofi^'s), by which latter is meant 
the "angle" between a circle and a tangent to it, since (Eucl. in. 16) the 
latter "angle" is less than any rectilineal angle whatever. The objection, it 
will be observed, assumes that the two sorts of angles are homogeneous. 
Plutarch and Carpus are classed among those who, in one way or other, placed 
an angle among magnitudes-^ and, as above noted, Plutarch claimed Apollonius 
as a supporter of his view, although the word contraction \o{ a surface or solid) 

I used by the latter does not in itself suggest magnitude much more than Euclid's 
I inclination. It was this last consideration which doubtless led " Aganis," the 
; " friend " (socius) apparently of Simplicius, to substitute for Apollonius' 
j wording " a quantity which has dimensions and the extremities of which arrive 
at one point'' (an-Nairizi, p. 13). 

2. Eudemus the Peripatetic, who wrote a whole work on the angle, main- 
tained that it belonged to the category of quality. Aristotle had given as his 
fourth variety of quality "figure and the shape subsisting in each thincj. and, 

H. £. 12 



I 



17^ BOOK I [l Dcfp. 8, 9 

besides these, straightness, curvature, and the like** (CMig^ries 8» 10 a 11). 
He says that each individual thing is spoken of as fuaie in respect of its fbmiy 
and he instances a triangle and a square, using them again later on (ifiU, 1 1 a 5) 
to show that it is not all qualities which are susceptible of man and less ; again, 
in Physics i. 5,, 188 a 25 a«^, strai^^ drcmlar are called kinds fA fy^re. 
Aristotle would no doubt have regarded deJUciwn (ffdcXdEirtfai) as belonging to 
the same category with straightness and curvature (fcofuraXoniv). At all eventi, 
Eudemus took up an angle as having its origin in the hnakmg or i^bdhn 
(fcAocrcc) of lines : deflection, he aigued, was quality if straightness was, and that 
which has its origin in quality is itself quality. Objectors to this view argued 
thus. If an an^le be a quality Qrmanyc) like heat or cold, how can it be bisCMSted, 
say ? It can m fact be divided ; and, if things of which divisibility is an 
essential attribute are varieties of quantum and not qualities, an angle cannot 
be a quality. Further, the man and the kss are the appropriate attributes tA 
quality, not the equal and the unequal ; if therefore an ande were a quality, 
we should have to say of angles, not that one is greater and another smaller, 
but that one is more an angle and another less an ang^ and that two angles 
are not unequal but dissimilar (nSrofUHOi). As a matter of fiurt, we are tokf by 
Simplicius, 538, 21, on Arist Da eaala that those who brought the an(^ uiKter 
the category of quale did call equal angles similar angles ; and Aristotle 
himself speaks of similar angles in this sense in Da coda 296 b 20, 311 b 34. 

3. Euclid and all who cidled an angle an inclination are held by Sjrrianos 
to have classed it as a nlation (vpof rt). Yet Euclid certainly rqparded varies 
as magnitudes; this is clear both -from the earliest propositions deaSng 
specifically with angles, e.g. i. 9, 13, and also ^though in another way) from 
his describing an angle in the very next defimtion and always as camAumed 
(n€pi€xofuyri) by the two lines forming it (Simon, Euclid^ p. 28). 

Proclus (i.e. in this case Syrianus) adds that the truth lies between these 
three views. The angle partakes in fact of all those categories: it needs the 
quantity involved in magnitude, thereby becoming susceptible of equality, 
inequality and the like ; it needs the quality given it by its form^ and lastly 
the nlation subsisting between the lines or planes bounding it 

Ancient classification of '* angles." 

An elaborate classification of angles given by Proclus (pp. 126, 7 — 127, 16) 
may safely be attributed to (jeminus. In order to show it by a diagram it I 

Angles 



on sarfaces in solids 
I {iv OTcpewt) 



r- 



on simpU surfaces on mixed surfaces 

I (e.g. cones, cylinders) 

I ' 1 

on phtus on spherical surfaces 

made by ntuple lines made by **mixid*^ lines bjr one of each 

(e.g. the angle made by a (e.g. the angle forAied by an 

curve, such as the cissM ellipse and its axis or by 

and kippapede, with itself) an ellipse and a drde) 

line-line line-drcumf. drcumf.-circumf. 

1 1 1 . H ,, 

Ime-convcx Ime-concave convex-convex concave-concave ikiued, or 

(eg. an^e of a e.g. horn-like (dfu^vproi) {dft^oiKM) convex-concave 

semicircle) {xtparoti^t) or ••scraper-like" (e.g. those of 

((MTporcdfcf) Utnes) 



1. Deff. 8, 9] NOTE. ON DEFINITIONS 8, 9 179 

will be necessary to make a convention about terms. Angles are to be under- 
stood under each class, " line-circumference *' means an angle contained by a 
straight line and an arc of a circle, " line-convex " an angle contained by a 
straight line and a circular arc with convexity outwards^ and so on in every 
case. 

Definitions of angle classified. 

As for the point, straight line, and plane, so for the angU^ Schotten gives 
a valuable summary, classification and criticism of the different modem views 
up to date (Inhalt und Methode des planimetrischen Unterrichts^ 11., 1893, 
pp. 94 — 183); and for later developments represented by Veronese reference 
I may be made to the second article (by Amaldi) in Questioni riguardanti 
\ la geometria eUmentare (Bologna, 1900) already referred to. 
I With one or two exceptions, says Schotten, the definitions of an angle may 

f be classed in three groups representing generally the following views : 
\ ' I. The angle is the difference of direction between tUH> straight lines. (With 
JL this group may be compared Euclid's definition of an angle as an inclination.) 
I 2. The angle is the quantity or amount (or the measure) of the rotation 
\ necessary to bring one of its sides from its own position to that of the other side 
without its moving out of the plane containing both, 

3. The angle is the 'portion of a plane included between ttvo straight lines in 
the plane which meet in a point {or two rays issuing from the point). 

It is remarkable however that nearly all of the text-books which give 
definitions different from those in group 2 add to them something pointing to 
a connexion between an angle and rotation : a striking indication that the 
essential nature of an angle is closely connected with rotation, and that a good 
definition must take account of that connexion. 

The definitions in the first group must be admitted to be tautologous, or 

circular^ inasmuch as they really presuppose some conception of an angle. 

Direction (as between two given points) may no doubt be regarded as a primary 

k notion; and it may be defined as "the immediate relation of two points which 

} the ray enables us to realise'' (Schotten). But ''a direction is no intensive 

^ magnitude, and therefore two directions cannot have any quantitative 

I difference" (Biirklen). Nor is direction susceptible of differences such as 

'those between qualities, e.g. colours. Direction is a jiVi^ar' entity: there 

cannot be different sorts or degrees of direction. If we speak of "a different 

direction," we use the word equivocally ; what we mean is simply "another" 

direction. The fact is that these definitions of an angle as a difference of 

direction unconsciously appeal to something outside the notion of direction 

. altogether, to some conception equivalent to that of the angle itself. 

Recent Italian views. 

I The second group of definitions are (says Amaldi) based on the idea of the 
rotation of a straight line or ray in a plane about a point : an idea which, 
I logically formulated, may lead to a convenient method of introducing the 
I angle. But it must be made independent of metric conceptions, or of the 
conception of congruence^ so as to bring out first the notion of an angle, and 
afterwards the notion of ^^wa/ angles. 

The third group of definitions satisfy the condition of not including metric 
conceptions ; but they do not entirely correspond to our intuitive conception 
of an angle, to which we attribute the character of an entity in one dimension 
i(as Veronese says) with respect to the ray as element, or an entity in tivo 






12- 



iSo BOOK I [l Dsr. 9 

dimensions with reference Xo points as elementSi which may be called an attgtdar 
sector. The defect is however easily remedied by considering the an^ as 
" the aggr^te of the rays issuing from the vertex and comprised in the angular . 
sector." 

Proceeding to consider the principal methods of arriving at the logical 
formulation of the first superficial properties of the ftam from whidi a 
definition of the angle may emerge, Amaldi distinguishes two points of view 
(1) XhtgtfutiCy (2) the actual. 

(i) From the first point of view we consider the dmster of straight Mnes 
or rays (the aggregate of all the straight lines in a plane oassing throush a 
point, or of all the rays with their extremities in that point) as generatea by 
the movement of a straight line or ray in the plane, about a point This leads 
to the postulation of a closed order^ or circular disposition^ of the straight lines 
or rays in a cluster. Next comes the connodon subsisting between the 
disposition of any two clusters whatever in one plane, and so oa 

(2) Starting from the point of view of the aitual^ we lay the foundation 
of the definition of an angle in the dimmn of the plane into two parts pialf- 
planes) by the straight line. Next, two straight lines (tf, i) in the plane, inter- 
secting at a point O^ divide the plane into four regions which are called 
angular sectors (convex) ; and finally the an^e (ab) or (Ai) may be defined as 
the aggregate of the rays issuing firam O CMd belonging to the angular sector 
which has a and hfor sides. 

Veronese's procedure (in his Elementi) is as follows. He bq;ins with the 
first properties of the plane introduced by^ the following definition. 

The figure given by all l^e straight lines joining Uie pomts of a straight 
line r to a point /' outside it and by 
the parallel to r through /' is called a 
cluster of straight lineSy a cluster of rays^ 
or a plane^ according as we consider 




the element of the figure itself to be the ^^y"^ j 

straight line^ the ray terminated at F^ P ^ 

or z, point. I 

[It will be observed that this method of producing a plane involves using { 
the parallel to r. This presents no difficulty to Veronese because he has 4 
previously defined parallels, without reference to the plane, by means of reflex 
or opposite figures, with respect to a point O : '' two straight lines are csdled < 
parallel^ if one of them contains two points opposite to (or the reflex of) two 
points of the other with respect to the middle point of a common transversal 
(of the two lines)." He proves by means of a postulate that the parallel r 
does belong to the plane Pr. Ingrami avoids, the use of the parallel by 
defining a plane as "the figure formed by the half straight lines which project ( 
'fi-om an internal point of a triangle (i.e. a point on a line joining any vertex of 
a three-side to a point of the opposite side) the points of its perimeter," and " 
then defining a cluster of rays as "the aggregate of the half straight lines in a 
plane ^starting from a given point of the plane and passing through the points I 
of the perimeter of a triangle containing the point."] J 

Veronese goes on to the definition of an angle. " We call an angle apart 
of a duster o/rays^ bounded by two rays (as the segment is a part of a straight ' 
line bounded by two points). 

^*An angle of the cluster^ the bounding rays of which are opposite^ is caUod a 
flat angle." 

Then, after a postulate corresponding to postulates which he lays down for 



I. Deff. 9-12] NOTES ON DEFINITIONS 9—12 181 

a rectilineal segment and for a straight line^ Veronese proves that all fiat angles 
are equal to one another. 



Uf9 




Hence he concludes that "the duster of rays is a homogeneous linear 
system in which the element is the ray instead of the point. The cluster 
being a homogeneous linear system, all the propositions deduced from 
[Veronese's] Post i for the straight line apply to it, e.g. that relative to 
the sum and difference of the segments : it is only necessary to substitute 
the ray for the point, and the angle for the segment.*' 

Definitions 10, 11, 12. 

10. *Orav 8c cMciOi lit cMcIaF orrotfcura ras ^^€^79 ytavta.% uras liXXi/Xaiv 
ir<k|J, ^ptfi; kKuripa toiv Tcraiv fa¥%mv inrif koI 17 ^^con^jcvia cMcui icd^croc fcoAcirai, 

1 1. *Afi)8Xcia yiavia iariv 1} /ittltop 6pBrj^ 

12. *Ofcui Sk ff IKaaatav 6p$7Js. 

10. fVhen a straight line set up on a straight line makes the adjacent angles 
equal to one anothery each of the equal angles is right, and the straight line 
stcmding on the other is called a perpendicular to that on which it stands, 

I II. An obtuse angle is an angle greater than a right angle. 

\ 1 2. An acute angle is an angle less than a right angle, 

\ ^^€^ is the regular term for adjacent angles, meaning literally " (next) in 

I ' order.'' . I do not find the term used in Aristotle of angles^ but he explains its 
i meaning in such passages as Physics vi. i, 231 b 8 : "those things are (next) 
i in order which have nothing of the same kind {crvyytvk) between them." 
► fca^crof , perpendicular^ means literally let fall : the full expression is perpen- 

I dicular straight line, as we see from the enunciation of Eucl. i. 11, and the 
\ notion is that of a straight line let fall upon the surface of the earthy dLpiumb- 
[ line, Proclus (p. 283, 9) tells us that in ancient times the perpendicular was 
called gnomon-tvise (#cara yv«ifioFa), because the gnomon (an upright stick) wtt^ 
f set up at right angles to the horizon. 

The three kinds of angles are among the things which according to the 
Platonic Socrates {Republic vi. 510 c) the geometer assumes and argues from, 
declining to give any account of them because they are obvious. Aristotle 
discusses the priority of the right angle in comparison with the acute (Metaph. 
1084 b 7): in one way the right angle is prior, i.e. in being defined (^n 
Zpurraj) and by its notion (rf Xoy<{»), in another way the acute is prior, i.e. as 

I being a party and because the right angle is divided into acute angles ; the 
acute angle is prior as mattery the right angle in respect oi form\ cf. also 
Metaph, 1035 b 6, ''the notion of the right angle is not divided into 



Definition 14. 



182 BOOK I [i. Deff. 12-14 

that of an acute angle, but the reverse ; for, when defining an acute angle, 
you make use of the right angle.*' Proclus (p. 133, 15) observes that it is by 
the perpendicular that we measure the heights of figures, and that it is by 
reference to the right angle that we distinguish the other rectilineal angles, 
which are otherwise undistinguished the one from the other. 

The Aristotelian Problems (16, 4, 913 b ^6) contain an ex|xression perhaps 
worth quoting. The question discussed is why things which &11 on the 
ground and rebound mikt ''similar" an^es with thMuraice on both sides of | 
the point of impact; and it is observed that '^.die right angle is the Nmii ^ 
(ofMs) of the opposite angles," where however *' opposite " seems to mean, not 
''supplementary " (or acute and obtuse), but the equal angles made with the ^ 
surface on opposite sides of the perpendicular. 

Proclus, after his manner, remarks that the statement that an angle less 
than a right angle is acute is not true without qualification, for f i]| the kmihUke 
angle (between the circumference of a drde ai\d a tangent) is less than a 
right angle, since it is less than an atuU angle, but is not an acute angle, wbik 
(2) the "angle of a semicircle " (between the arc and a diameter) is also less 
than a right angle,* but is not an acute angle. % 

The existence of the right angle is X)f course proved in L 1 1. 

Definition 13. 

Opo% iariv, o tivos iari wipa$. 

A boundary is that which is an extremity rfemy thing. 

Aristotle also uses the words 2pof and t^s as synonymous. Cf. Drgm, 
animaL ii. 6, 745 a 6, 9, where in the expression "h'mit of magnitude** first 
one and then the other word is used. 

Proclus (p. 136, 8) remarks that the word boundary is appropriate to the 
origin of geometry, which began from the measurement of areas of ground 
and involved the marking of .boundaries. 



i 



Sx^fui loTi TO viro TIV09 17 rivwv op«iiv mpuxofityov, . 

A figure is that rvhich is contained by any boundary or boundaries. \ 

Plato in the Meno observes that roundness (trrpor^Xifnfi) or the round is a ^ 

" figufe," and that the straight and many other thmgs are so too ; he then j 

inquires what there is common to all of thejxi, in virtue of which we apply the | 

term "figure" to them. His answer is (76 a): "with reference to every j 
fi^re I say that that in "u^hich the solid terminates (rovro^ ci9 u rh oryMlw 



irtfHuv€C) is a figure^ or, to put it briefly, a figure is an extremity of a solid.** I 
The first observation is similar to Aristotle's in the Physics i. 5, 188 a 25, < 
where angle, straight, and circular are mentioned as genera of figure: In the 
Categories 8, 10 a 11, "figure" is placed with straighthess and curvedness in 
the category' of quality. Here however "figure" appears to mean skafe I 
(fiap^') rather than " figure " in oiir sense. Coming hearer to "figure" in our ! 
sense, Aristotle admits that figure is ""a sort of magnitude" i^De anima in. i, \ 
425 a 18X and he distinguishes plcme figures of two kinds, in language not 
unlike Euclid's, as contained by straight and circular lines re^)ectively : "every 
plane figure is either rectilined or formed by circular- lines (vcpt^cpoypa/ifuiy), 
and the rectilineal figure is contained by several lines, the circular by one 
line " {De caelo 11. 4, 286 b 13). He is careful to explain that a plane is not a 



I. Deff. 14-16] NOTES ON DEFINITIONS 12—16 183 

figure, nor a figure a plane, but that a plane figure constitutes one notion and 
is a species of the genus figure {AnaL post, n. 3, 90 b 37). Aristotle does not 
attempt to define figure in general, in fact he says it would be useless : " From 
this it is clear that there is one definition of soul in the same way as there is 
one definition oi figure \ for in the one case there is no figure except the 
triangle, quadrilateral, and so on, nor is there any soul other than those above 
mentioned. A definition might be constructed which should apply to all 
figures but not specially to any particular figure, and similarly with the 
species of soul referred to. [But such a general definition would serve no 
purpose.1 Hence it is absurd here as elsewhere to seek a general definition 
which will not be properly a definition of anything in existence and will not 
be applicable to the particular irreducible species before us, to the neglect of 
the definition which is so applicable" {De anima 11. 3, 414 b 20^ — 28). 

Comparing Euclid's definition with the above, we observe that by. intro- 
ducing boundary {opos) he. at once excludes the straight which Aristotle classed 
as figure ; he doubtless excluded angle also, as we may judge by (i) Heron's 
statement that ''neither one nor two straight lines can complete a figure," 
(3) the alternative definition of a straight line as *'that which cannot with 
another line of the same species form a figure," (3) Geminus' distinction 
between the line irhxcYi forms a figure (crxfffJLarowoiovaa) and the line which 
extends indefinitely {iw* aTrcipov iKpaXXxt/uvri), which latter term includes a 
hjrperbola and a parabola. Instead of calling figure an extremity as 
Plato did in the expression '^extremity (or limit) of a solid,** Euclid 
describes a figure as that which has a boundary or boundaries. And lastly, 
in spite of Aristotle's objection, ht does attempt a general definition to 
cover all kinds of figure, solid and plane. It appears certain therefore that 
Euclid's definition is entirely his own. 

Another view of a figure, recalling that of Plato in Meno 76 a, is attributed 
by Proclus (p. 143, 8) to Posidonius. The latter regarded the.^^r^ as the 
confining extremity or limit (wipas cnrvicXciov), "separating the notion of figure 
firom quantity (or magnitude) and making it the cause of definition^ limitation^ 
and inclusion (roO iipL<r$a%, icoi ir€W€pcur6<u ical t^ itc^mox^s)... Posidonius thus 
seems to have in view only the boundary placed round from outside, Euclid 
the whole content, so that Euclid will speak of the circle as a figure in 
respect of its whole plane (surface) and of its inclusion (firom) without, whereas 
Posidonius (makes it a figure) in respect of its circumference... Posidpnius 
wished to explain the notion of figure as itself limiting and confining magnitude." 

Proclus observes that a logical and refining critic might object to Euclid's 

definition as defining the genus from the species, since that which is enclosed 

by one boundary and that which * is enclosed by several are both species of 

figure. The best answer to this seems to be supplied by the passage of 

^ Aristotle's I>e anima quoted above. 

; Definitions ,15, 16. ^ 

I 15. KvfcXos iari <r}(rjfta ciriircSov vwo fiuis ypofifi^c mpuxofifvoy [^ fcaXcinu 

T^H^M^>cta|, wp^ ^¥ d^* M^ fnffJL€iov rwv cvros tov fr^mpxiro^ icccficvaiv n-curai at 
\ wpovwiirrownii cii^cuu {irpof rrjv rov kvkXov ir€pt/^€p€iay\ urcu dXAifXacc curiv. 
' 1 6. Kivrpw 8i rov.KVicXov ro airfp,€iO¥ fcaXcirac 

1$. A circle is a J>lane figure contained by one line such that all the straight 
lines falling upon it from one point among those lying within the figure are equal 
to one another ; 

16. And the point is called the ctnXxe of the circle. 



i84 BOOK I [i. Dbff. t5» i6 

The words ^ KoXctTai xcpc^^io, *' which is called the drcumferenoe," and 
wpoi rifv Tov kvkXov w€piitiip€ua»f "to the circuinference of the drde," aie 
bracketed by Heiberg because, although the Mss. have them, they aie 
omitted in other ancient sources, viz. Prmdus, Taurus, Sextus Empiricus and 
Boethius, and Heron also omits the second gloss. The recently discovered 
papyrus Herculanensis No. 1061 also quotes the definition without the words 
m question, confirming Heibeig's rejection of them (see Heiberg in Hermes 
XXXVIII., 1903, p. 47). The words were doubtless added in view of the 
occurrence of the word "circumference'' in Deff. 17, 18 immediatdv 
following, without any explanation. But no explanation was needed. Though 
the word xc/M^cpcia does not occur in Plato, Aristotle uses it seyeml timet 

( 1 ) in the general sense of contour without any special mathematical signification, 

(2) mathematically, with reference to the rainbow and the circumference, as 
well as an arc, of a circle. Hence Euclid was perfectly justified in emtdoying 
the word in Deff. 17, 18 and elsewhere, but leaiong it undefined as being a 
word universally understood and not involving in itself any mathematical 
conception. It may be added that an-NairlzI had not the bracketed words 
in his text ; for he comments on and tries to explain Euclid's omission to 
define the circumference. 

The definition itself contained nothing new in substance. Plato {Parme- 
nides 137 e) says : " Round is, I take it, that the extremes of which are every 
way equally distant from the middle " (irrpoyyvAor yc vov Icrrt ToOr«s oS & td 
laxora warraxj awo rav ficcrov lirov Awixff). In Aristotle we find the following 
expressions: "the circular (xfpi^€pdypo/4fu>v) plane figure bounded by one 
line" (Df caelo 11. 4, 286 b 13 — 16); "the plane equal (Le. extending equally 
all ways) from the middle" (IrArcSoK to lie rov /mctov \ffw\ meaning a 
circle (^Rhetoric iii. 6, 1407 b 37); he also contrasts with the circle ^'any 
other figure which has not the lines from the middle equal, as for example an 
egg-shaped figure" {De caelo ii. 4, 287 a 19). The word "centre" («€»^pw) 
was also regularly used : cf. Proclus* quotation from the "oracles" (Aoyia), 
" the centre from which all (lines extending) as far as the rim are equal." 

The definition as it stands has no genetic character. It says nothing as to 
the existence or non-existence of the thing defined or as to the method of 
constructing it. It simply explains what is meant by the word " circle," and 
is a provisional definition which cannot be used until the existence of circles 
is proved or assumed. Generally, in such a case, existence is proved by 
actual construction; but here the possibility of constructing the circle as 
defined, and consequently its existence, are /^j/^/17/^// (Postulate 3). A genetic 
definition might state that a circle is the figure described when a straight line, 
always remaining in one plane, moves about one extremity as a fixed point 
until it returns to its first position (so Heron, Def. 29). 

Simplicius indeed, who points out that the distance between the feet of a 
pair of compasses is a straight line from the centre to the circumference, will 
have it that Euclid intended by this definition to show how to construct a 
circle by the revolution of a straight line about one end as centre ; and an- 
Nairizi points to this as the explanation (i) of Euclid's definition of a circle 
as 2i plane figure^ meaning the whole surface bounded by the circumference, 
and not the circumference itself, and (2) of his omission to mention the 
"circumference," since with this construcrion the circumference is not drawn 
separately as a line. But it is not necessary to suppose that Eudid himself 
did more than follow the traditional view ; for the same conception of the 
circle as k plane figure appears, as we have seen, in Aristotle. While, however. 



If 



I. Deff. 15-17] NOTES ON DEFINITIONS 15—17 185 

Euclid is generally careful to say the ^^circumference of a circle " when he means 
the circumference, or an arc, only, there are cases where "circle" means 
"circumference of a circle," e.g. in in. 10: "A circle does not cut a circle 
in more points than two." 

Heron, Produs and Simplidus are all careful to point out that the centre 
is not the only point which is equidistant from all points of the circumference. 
The centre is the only point in the plane of the circle ("lying within the figure," 
as Euclid says) of which this is true; any point not in the same plane which 
is equidistant from all points of the circumference is a pote. If you set up a 
"gnomon " (an upright stick) at the centre of a circle (i.e. a line through the 
centre perpendicular to the plane of the circle), its upper extremity is a pole 
(Proclus, p. 153, 3); the perpendicular is the locus of all such poles. 

Definition 17. 

Ataficr/>09 8c rov kvkXov ktrrXv cMcui ric 81a rw Kiirrpov rjyitiyti ical irtparov- 
ficn; i^* hcaripa ra /upij vro r^^ rev kvkXov w€pif^€p€ias, i/ric icoi &\a tc/avci rov 

JCVfcAoF. 

A diameter of the circle is any straight lifie drawn through the centre and 
terminated in both directions by the circumference of the circle^ and such a ^reught 
line also bisects the circle. 

The last words, literally "which (straight line) also bisects the circle," 
are omitted by Simson and the editors who followed him. But they are 
necessary even though they do not "belong to the definition" but only 
express a property of the diameter as defined. For, without this explanation, 
Euclid would not have been justified in describing as a j^/rrAnrcle a portion 
of a circle bounded by a diameter and the circumference cut off by it 

Simplicius observes that the diameter is so called because it passes through 
the whole surface of a circle as if measuring it, and also because it divides the 
circle into two equal paits. He might however have added that, in general, it 
is a line passing through a figure where it is widest^ as well as dividing it 
equally: thus in Aristotle ra #cara huL/urpoy icct/Acva, "things diametrically 
situated ** in space, are at their maximum distance apart Diameter was the 
regular word in Euclid and elsewhere for the diameter of a square^ and also 
of a parallelogram; diagonal (Siayuvios) was a later term, defined by Heron 
(Def. 68) as the straight line drawn from an angle to an angle. 

Proclus (p. 157, 10) says that Thales was the first to prove that a circle is 
bisected by its diameter; but we are not told how he proved it Proclus gives 
as the reason of the property " the undeviating course of the straight line 
through the centre " (a simple appeal to symmetry), but adds that, if it is 
desired to prove it mathematically, it is only necessary to imagine the diameter 
drawn and one part of the circle applied to the other ; it is then clear that 
they must coincide, for, if they did not, and one fell inside or outside the 
other, the straight lines from the centre to the circumference would not all be 
equal : which is absurd. 

Saccheri's proof is worth quoting. It depends on three "Lemmas" 
immediately preceding, (i) that two straight lines cannot enclose a space, 

!2^ that two straight lines cannot have one and the same segment common, 
3) that, if two straight lines meet at a point, they do not touch, but cut one 
another, at it. 

" Let MDHNKM be a circle, A its centre, MN a diameter. Suppose 



i 





i86 BOOK I [i. Dkff. 17, 

the portion MNKM of the circle turned about the fixed points M^ N^ 90 
that it ultimately comes near to or coincides with the remaining portion 
MNHDM. 

"Then (i) the whole diameter MAl^^ with all 
its points, clearly remains in the same position, 
since otherwise two straight lines would enclose a 
space (contrary to the first Lemma). 

'' (ii) Clearly no point K of the circumference 
NKM falls withm or outside the surfieure enclosed 

by the diameter if^A^and the other part, JV»Z>i/; v \/ J 

of the circumference, since otherwise, contrary to N. ^x M 

the nature of the circle, a radius as AK would be ^"^ --^ I 

less or ^eater than another radius as AH. \ 

*' (iii) Any radius MA can clearly be rectilineally produced only along a 
single other radius AN^ since otherwise (contrary to the second Lemma) two 
lines assumed straight, e.g. MAN^ MAH^ would have one and the same ^ 
common segment 

" (iv) All diameters of the circle obviously cut one another in the centre 
(Lemma 3 preceding), and they bisect one another there, by the go^eral 
properties of the circle. 

" From all this it is manifest that the diameter MAN divides its circle 
and the circumference of it just exactly into two equal parts, and the same 
ma^ be generally asserted for every diameter whatsoever of the same circle ; 
which was to be proved." 

Simson observes that the property is easily deduced from in. 31 and 24 ; 
for it follows from iii. 31 that toe two parts of the circle are ^similar 
segments" of a circle (segments containing equal angles, in. Def. 11), and 
from III. 24 that they are equal to one another. 

Definition 18. 

'HfUJcvicXuiy Sc Itm ro irtpvf^iktvov vyri^a. ^ro re rSjs Sta/icrpov fcot r^ , 
dxoAafi)3ako/Mn^ vtt* aMj^ ir€pi/^€p€ia%, Kivrpov 8c rov iffiucvKkiov r^ at^ i I 
icoi rov kvkXov IcrrtV. y 

A semicircle is the figure contained by the diameter and the circumference cut 
off by it. And the centre of the semicircle is the same as that of the circle. 

The last words, '*And the centre of the semicircle is the same as that ( 
of the circle," are added from Proclus to the definition as it appears in the 
Mss. Scarburgh remarks that a semicircle has no centre, properly speaking, | 
and thinks that the words are not Euclid's, but only a note by Proclus. I am \ 
however inclined to think that they are genuine, if only because of the very * 
futility of an observation added by Proclus. He explains, namely, that the ' 
semicircle is the only plane figure that has its centre on its perimeter (!), "so 
that you may conclude that the centre has three positions, since it may be i 
within the figure, as in the case of a circle, or on the perimeter, as with the I 
semicircle, or outside, as with some conic Unes (the single-branch hyperbola ' 
presumably)" ! > 

Proclus and Simplicius point out that, in the order adopted by Euclid for 
these definitions of figures, the first figure taken is that bounded by one line 
(the circle), then follows that bounded by two lines (the semicircle), then the 
triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes 



1 



:v 



1. Deff. 18-21] NOTES ON DEFINITIONS 17—21 187 

different kinds of figures bounded by two lines (pp. 159, 14 — 160, 9). Thus 
they may be formed 

(i) by circumference and circumference, e.g. (a) those forming angles, as 
a /un^ (to fArfvo€t&€^) and the figure included by two arcs with convexities 
outward, and (d) the angle-less (dycJi^iOF), as the figure included between two 
concentric circles (the coronal) \ 

(2) by circumference and straight line, e.g. the semicircle or segments of 
circles (a^iScc is a name given to those less than a semicircle); 

(3) by "mixed" line and "mixed" line, e.g. two ellipses cutting one 
another; 

(4) by " mixed " line and circumference, e.g. intersecting ellipse and 
circle ; 

(5) by " mixed " line and straight line, e.g. half an ellipse. 

Following Def. 18 in the mss. is a definition of a segment of a circle which 
was obviously interpolated from 111. Def. 6. Proclus, Martianus Capella and 
Boethius do not give it in this place, and it is therefore properly omitted. 

Definitions 19, 20, 21. 

19. Sx^fuira tifOvypafAfia ioTi rk vro cii^cuSv ir^icxoficva, rpiVXcvpa /acv 
ra viro rprnVf rcrpairXcvpa Sc ra {fwo rc<r<rapci>v, woKvvkwpa 8c ra vro nk^iovtav ij 
Ttavapiav €vOtiwv ir€pi€XPfi€V€L 

20. Toiv &k rpivkwpmv <ryyiimwv Urivktvpov fiky Tpiyuiv6y iari ro ra« rpcts 
urac ixpv xXcvpac, uroa'K€kU Sk to rac fivo fi6yas Icras Ix^vk vXcvpav, (ffcaXiTvov Sk 
TO Ttts Tpcis dvurov^ l)(ov wktvpas, -^ 

21. *ETi Sc TOIV TpiirXcvp<Dv <r)(iffMariav dpOaytivtov /acv rplywyop Icrri to c^ov 
6fArjv ywvlavy dfiPkvyiaviov 8c to I^of d/ifiktZay ycDFtiav, 6$vyiivtoy 8c to Tas TpciS 

19. Rectilineal figures are those which are contained by straight lines^ 
trilateral figures being those contained by three^ quadrilateral those contained by 

four^ and multilateral those contained by more than four straight lines, 

20. Of trilateral figures, an equilateral triangle fx that which has its three 
sides equals an isosceles triangle that which has two of its sides alone equcU^ and 
a scalene triangle thcU which has its three sides unequal. 

2 1. Further^ of trilateral figures^ a right-angled triangle is that which has 
a right angle^ an obtuse-angled triangle that which has an obtuse angle^ and an 
acute-angled triangle that which has its three angles acute, 

19. 
The latter part of this definition, distinguishing three-sided^ four-sided and 
many-sided figures, is probably due to Euclid himself, since the words 
TpiirXcvpov, TCTpan-Xcvpov and xoXvVXcvpok do not appear in Plato or Aristotle 
(only in one passage of the Mechanics and of the Problems respectively does 
even rcTpaVXcupov, quadrilateral^ occur). By his use of TCTpaVXcvpov, 
quadrilateral, Euclid seems practically to have put an end to any ambiguity 
in the use by mathematicians of the word TCTpayuvov, literally "four-angled 
(figure)," and to have got it restricted to the square. Cf. note on Def. 22. 

20. 

Isosceles {uroaictkiii, with equal legs) is used by Plato as well as Aristotle. 
Scalene (<rKakr/y6% with the varient frKakrpnj^) is used by Aristotle of a triangle 
with no two sides equal: cf. also Tim. Locr. 98 b. Plato, Euthyphro 12 d. 



\ 




i88 BOOK I [t DiPP. so, sa 

applies the term '' scalene " to an odd number in contrast to ^ isosceles " used 
of an even number. Proclus (p. i68, 24) seems to connect it with 9taiff^ to 
limp ; others make it akin to irmAi^ aroohedy aslani. Apollonius uses the 
same word " scalene '* of an obUpti circular cone. 

Triangles are classified, first with reference to their sides, and then with 
reference to their angles. Produs points out that seven distinct species of 
triangles emerge: (i) the equiiiUirai triangle, (a) three species of isasteies 
triangles, the r^;ht-angled, the obtuse-angM and the acute-angled, (3) the 
same three' varieties of scalene triangles. 

Proclus gives an odd reason for the dual classification according to sides 
and angles, namely that Euclid was mindful of the ftct that it is not every 
triangle that is trilateral alsa He explains this statement by reference 
(p. 165, 22) to a figure which some called barb4iki (d«ctotiSiyv) while 
Zenodorus odled it koUauhangkd (MocXoyvJMoc). Proclus mentions it again 
in his note on i. 22 (p. 328, 21 sqq[.) as one 61 the paradoxes of f;eoinetr]|r, 
observing that it is seen in the figure of that proposition. This "triangle" is 
merely a quadrilateral with a re-entrant angle ; and the idea that 
it has only three angles is due to the n(m-recognition of the 
fourth angle (which is greater than two right angles) as being an 
angle at all. Since Proclus speaks of \h<t fomr-sided trian^ as 
''one of the paradoxes in geometry,'' it is perhaps not sdfe to 
assume that the misconception underlying tl^ expression existed 
in the mind of Proclus alone ; but there does not seem to be any evidence 
that Zenodorus called the figure in question a triangle (cf. Pappus, ed. 
Hultsch, pp. 1 1 54, 1206). I 

Definition 23. j 

Tcav 8i rcrpaxXcvpuiv axflfdrtav rtrpaywvov fiiv loriv, i UrAwXoffi^w ri ion I 
fcoi 6p0oywtoVf Ircpofii^fccs &c^ i 6p6oyw¥iov fih^^ oIk uFoirXcvpov hiy ^fLpo9 8c, i I 
laowXwfiw /livf cWmc 6p6oyw¥iO¥ 8^ (&o/aJSoci8<s Sk ro r&s Airtyavrtov wkwvpai re irai i 
ytiwia,^ uras dXAi^Acus ^XP^i ^ ^^ laowXtvpov Itrrw ovt€ 6pOoyiayioy ra 8i wapa 
ravra rerpdrrXtvpa rpa7rc(ia KaXturdm, I 

Of quadrilateral figures^ a square is that which is both equilateral and right- k 
angled; an oblong that which is right-angled but not equilateral; a rhombus \ 
that which is equilateral but not right-an^d; and a rhomboid that which has ^ 
its opposite sides and angles equal to one another but is neither equilateral nor \ 
right-angUd. And let quadrilaterals other than these be called trapezia. . 

rcrfMywrov was already a square with the Pythagoreans (cf. Aristotle, 
Metaph, 986 a 26), and it is so most commonly in Aristotle ; but in De anima < 
II. 3, 414 b 31 it seems to be a quadraateral, and in Metaph, 1054 b 2, ( 
'' equal and equiangular rcrpayoiva," it cannot be anything else but quadri- ' 
lateral if "equiangular" is to have any sense. Though, by introducing { 
TtrpanrXitvpw for any quadrilateral, Euclid enabled ambiguity to be avoided, 
there seem to be traces of the older vague use of rcrpayuiov in much later 
writers. Thus Heron (Def. 104) speaks of a cube as "contained by six equi- I 
lateral and equiangular r^rpaytava" and Proclus (p. 166, 10) adds to his ! 
remark about the " four-sided triangle ** that " you might have ra-paytava with 
more than the four sides," where TtTpdytova can hardly mean squares. ' 

Ircpdfii^iccf, oblong (with sides of different length)^ is also a Pythagorean term. 

The word right-angled (^oywiov) as here applied to quadrilaterals 
must mean rectangular (Le., practically, having all its angles right angles); 
for, although it is tempting to take the word in the same sense for a 



i 

i 






I. Def. 22] NOTES ON DEFINITIONS 20—22 189 

square as for a triangle (i.e. '* having one right angle "), this will not do in the 
case of the oblong, which, unless it were stated that (Aree of its angles are 
right angles, would not be sufficiently defined. 

If it be objected, as it was by Todhunter for example, that the definition 
of a square assumes more than is necessary, since it is sufficient that, being 
equilateral, it should have one right angle, the answer is that, as in other cases, 
the superfluity does not matter from Euclid's point of view ; on the contrary, 
the more of the essential attributes of a thing that could be included in its 
definition the better, provided that the existence of the thing defined and its 
possession of all those attributes is proved before the definition is actually 
used ; and Euclid does this in the case of the square by construction in i. 46, 
making no use of the definition before that proposition. 

The word rfunnbus (po/i^os) is apparently derived from ^cfijSoi^ to turn 
roufid and round^ and meant among other things a spinning-top. Archimedes 
uses the term solid rhombus to denote a solid figure made up of two right 
cones with a common circular base and vertices turned in opposite directions. 
We can of course easily imagine this solid generated by spinning \ and, if the 
cones were equal, the section through the common axis would be a plane 
rhombus, which would also be the apparent form of the spinning solid to the 
eye. The difficulty in the way of supposing the plane figure to have been 
named after the solid figure is that in Archimedes the cones forming the solid 
are not necessarily equal. It is however possible that the solid to which the 
name was originally given was made up of two equal cones, that the plane 
rhombus then received its name from that solid, and that Archimedes, in 
taking up the old name again, extended its signification (cf. J. H. T. Miiller, 
Beitrdge zur Terminologie der griechischen Mathematiker^ i860, p. 20). 
Proclus, while he speaks of a rhombus as being like a shaken, Le. deformed, 
square, and of a rhomboid as an oblong that has been moved, tries to explain 
the rhombus by reference to the appearance of a spinning square {r^pdyw^v 

jk}flpOVfL€VOv). 

It is true that the definition of a rhomboid says more than is necessary in 
describing it as having its opposite sides and angles equal to one another. 
The answer to the objection is the same as the answer to the similar objection 
to the definition of a square. 

Euclid makes no use in the Elements of the oblongs the rhombus^ the 
rhomboid^ and the trapezium. The explanation of his inclusion of definitions 
of the first three is no doubt that they were taken from earlier text-books. 
From the words ^'let quadrilaterals other than these be called trapezia," we 
may perhaps infer that this was a new name or a new application of an old 
name. 

As Euclid has not yet defined parallel lines and does not anywhere 
define a parallelogram^ he is not in a position to make the more elaborate 
classification of quadrilaterals attributed by Proclus to Posidonius and 
appearing also in Heron's Definitions. It may be shown by the following 
diagram, distinguishing seven species of quadnlaterals. 

Quadrilaterals 



parallelograms non-parallelograms 



rectangular non-rectangular two sides parallel no sides parallel 

I I (trapetium) (trapfioi4i) 

I — ' — I I • 1 I ' 1 

square pbhn^ rhombus rhotubcid isosceles trapetium ualeuo trapeuum 



I 



I90 BOOK I [i. Deff. 33, S3 

It will be observed that^ while Euclid in the above definition classes as 
trapezia all quadrilaterals other than iquares, obl(»igi, rfaombi, and rhomboid^ 
the word is in this classification restncted to quadrilaterals having two sides 
(only) parallel, and trapezoid is used to denote the rest £uclid appears to 
have used trapezium in the restricted sense of a quadrilateral with two sides 
parallel in his book xc^m ScoipccrcMr (on divisions of figures). Archimedes 
uses it in the same sense, but in one place describes it more precisely as a 
trapezium with its two sides parallel. 

Definition 23. 

napoAXiyAoi clcru^ tiSuax^ oTrtvcf If ny uifT^ hr a r A ff oSooi fcat Ik^oXAi^^mmu 
CIS air€tpov 1^* kicdrtpa ra yuifpii hn fiifiirMfio. uvf/ariwrowrw AAAijXaif. 

Parallel straight tines are straight Ufus whieh^ being in the same plane emd 
being produced indefinitely in bath Unctions^ do not meet one another in either 
dirution, 

IlapaAAi^Xof (alongside one another) written in one word does not appear 
in Plato ; but with Aristotle it was already a familiar term. 

€h aircifK>F cannot be translated ''to infinity" because these words might 
seem to suggest a r^n or place infinitely distant, whereas ck avcipor, which 
seems to be used indifferently with hr mipoy, is adverbial, meaning ''without 
limit," i.e. *' indefinitely." Thus the expr^sion is used of a magnitude beiog 
"infinitely divisible," or of a series of terms extending without limit 

In both directions^ c^* indrtfia, ra §i4pi^ literally *' towards both the parts" 
where *'parts" must be used in the sense of "regions" (cf Thuc ii. 96). 

It is clear that with Aristotle the general notion of parallels was that of 
straight lines which do not meet^ as in Euclid : thus Aristotle discusses the 
question whether to think that parallels do meet should be called a 
geometrical or an ungeometrical error {Anal, post, i. 12, 77 b 22), and (more 
interesting still in relation to Euclid) he observes that there is. nothing 
surprising in different hypotheses leading to the same error, as one might 
conclude that parallels meet by starting from the assumption, either (a) that 
the interior (angle) is greater than the exterior, or (b) that the angles of a 
triangle make up more than two right angles (Anal, prior. 11. 17, 66 a 11). 

Ajiother definition is attributed by Proclus to Posidonius, who said that 
^^ parallel lines are tlwse which^ {being) in one plane^ neither converge nor diverge^ 
but have all the perpendiculars equal which are drawn from the points of one 
line to the other ^ while such (straight lines) as make the perpendiculars less and 
less continually do converge to one another ; for the perpendicular is enough 
to define (opt^ciF hdvaraC) the heights of areas and the distances between lines. 
For this reason, when the perpendiculars are equal, the distances between the 
straight lines are equals but when they become greater and less, the interval is 
lessened, and the straight lines converge to one another in the direction in 
which the less perpendiculars are" (Proclus, p. 176, 6—17). 

Posidonius' definition, with the explanation as to distances between straight { 
lines, their convergence and divergence, amounts to the definition quoted by ! 
Simplidus (an-Nairizi, p. 25, ed. Curtze) which described straight lines as | 
parallel if when they are produced indefinitely both ways^ the distance between 
themy or the perpendicular drawn from either of them to the other ^ is always 
equal and not different. To the objection that it should be prorved that the 
distance between two parallel lines is the perpendicular to them Simplicius 



•^1 



1. Def. 23] NOTES ON DEFINITIONS 22, 23 191 

replies that the definition will do equally well if all mention of the perpen- 
dicuiar be omitted and it be merely stated that the distance remains equal, 
although " for proving the matter in Question it is necessary to say that one 
straight line is perpendicular to both'' (an-Nairizi, ed. Besthom-Heiberg, p. 9). 
He then quotes the definition of "the philosopher Aganis": ^^Parailei 
straight Ones are straight lines^ situated in the same plane^ the distance between 
whichj if they are produced iiidefiniidy in both directions at the same time^ is 
everywhere the sameJ' (This definition forms the basis of the attempt of 
"Aganis" to prove the Postulate of Parallels.) On the definition Simplicius 
remarks that the words ''situated in the same plane'' are perhaps unnecessary, 
•since, if the distance between the lines is eve^where the same, and one does 
not incline at all towards the other, they must for that reason be in the same 
plane. He adds that the ''distance" referred to in the definition is the 
shortest line which joins things disjoined.. Thus, between point and point, 
the distance is the straight line joining them ; between a point and a straight 
line or between a point and a plane it is the perpendicular drawn from the pomt 
to the line or plane; "as regards the distance between two lines, that distance 
b, if the lines are parallel, one and the same, equal to itself at all places on 
the lines, it is the shortest distance and, at all places on the lines, perpendicular 
to both " (i^k/. p. 10). 

The same idea occurs in a quotation by Proclus (p. 177, 11) from 
Geminus. As part of a classification of lines which do not meet he observes : 
" Of lines which do not meet, some are in one plane with one another, others 
not. Of those which meet and are in one plane, sofne are always the same 
distance from one another^ others lessen the distance continually, as the hyper- 
bola (approaches) the straight line, and the conchoid the straight line (i.e. the 
asymptote in each case). For these, while the distance is being continually 
lessened, are continually (in the position oQ not meeting, though they converge 
to one another ; they never converge entirely, and this is the most paradoxical 
theorem in geometry, since it shows that the convergence of some lines is non- 
convergent. But of lines which are always an equal distance apart, those 
which are straight and never make the (distance) between them smaller, and 
which are in one plane, are parallel." 

Thus the eguidistance-thieory of parallels (to which we shall return) is very 
fully represented in antiquity. I seem also to see traces in Greek writers of a 
conception equivalent to the vicious directian-Htieory which has been adopted 
in so many modem text-books. Aristotle has an interesting, though obscure, 
allusion in Ana/, pricr. 11. 16, 65 a 4 to a, petitio principii committed by "those 
who think that diey draw parallels " (or " establish the theory of parallels," 
which is a possible translation of ra« ira^KiXAi^Xov« ypa^civ): "for they un- 
consciously assume such things as it is not possible to demonstrate if parallels 
do not exist" It is clear from this that there was a vicious circle in the then 
current theory of parallels ; something which depended for its truth on the 
properties of parallels was assumed in the actual proof of those properties, 
e.g. that the three angles of a triangle make up two right angles. This is not 
the case in Euclid, and the passage makes it clear that it was Euclid himself 
who got rid of the petitio principii in earlier text-books by formulating and 
premising before i. 39 the famous Postulate 5^ which must ever be r^arded 
as among the most epoch-making achievements in the domain of geometry. 
But one of the commentators on Aristotle, Philoponus, has a note on the 
above passage purporting to give the specific character of the petitio principii 
alluded to; and it is here that a direction-theory of parallels may be hinted at. 



192 BOOK I [i. DBr. 13 

whether Philoponus is or is not rig^t in supposing Aat this was what Aristotle 
had in mind. Philoponus says: ''The same thinff is done by those iriio dnw 
parallels, namely begging the original question; for thc^ will have it tfiat it is 
possible to draw parallel straight lines from the mendian circle, and they 
assume a point, so to say, falluiff on the plane of that circle and thus they 
draw the straight lines. And what was sought is thereby assumed; for he 
who does not admit the genesis of the parallels will not admit the point 
referred to either." What is meant is, I think, somewhat as follows. Given 
a straight line and a point through which a parallel to it is to be drawn, we 
are to suppose the given straight line placed in the plane of the meridian. J 
Then we are told to draw through the given point anodier straight line in the ^ 
plane of the meridian (strictly speaking it should be drawn in a plane pualld k 
to the plane of the meridian, but the idea is that, compared with the sixe of | 
the meridian circle, the distance between the point and the straight line is { 
negligible) ; and this, as I read Philoponus, is supposed to be equivalent to j 
assuming a very distant point in the meridian plane and joining the given J 
point to it But obviously no ruler would stretch to such a pomt, and the 
objector would say that we cannot really direct a straight line to die assumed 
distant point except by drawing it, widiout more ado, paralld to the given 
straight line. And herein is the pttiHo frmcifii, I am confirmed in seeing 
in Philoponus an allusion to a dineiiOH'thearj by a remark of Schotten on a 
similar reference to the meridian plane supposed to be used by advocates of . 
that theory. Schotten is aiguing that direction is not in itself a conception ' 
such that you can predicate Me direction of two different lines. ** If any one 
should reply that nevertheless many lines can be conceived which aU have the 
direction from north to south^^ he replies that this represents only a nominal, 
not a real, identity of direction. 

Coming now to modem times, we may classify under three groups 
practically all the different definitions that have been given of parallels 
(Schotten, op, cit. 11. p. 188 sqq.). 

(i) Parallel straight lines hm*e no point common^ under which general 
conception the following varieties of statement may be included : 

{a) they do not cut one another^ ^ 

{b) they meet at infinity^ or ^ 

(c) they have a common point at infinity. \ 

(2) Parallel straight lines have the same^ or lihe^ direction or directions^ 
under which class of definitions must be included all those which introduce ' 
transversals and say that the parallels mahe equal angles with a transverscU. . 

(3) Parallel straight lines have the distance t^etween thetn constant \ \ 
with which group we may connect the attempt to explain a parallel as the > 
geometrical locus of all points which are equidistant from a straigkt line. { 

But the three points of view have a good deal in common ; some of them 
lead easily to the others. Thus the idea of the lines having no point common 
led to the notion of their having a common point at infinity, through the | 
influence of modem geometry seeking to embrace different cases under one f 
conception ; and then again the idea of the lines having a common point at ^ 
infinity might suggest their having the same direction. The " non-secant " f 
idea would also naturally lead to that of equidistance (3), since our 
observation shows that it is things which come nearer to one another that 
tend to meet, and hence, if lines are not to meet, the obvious thing is to see 
that they shall not come nearer, i.e. shall remain the same distance apart. 



\ 



I 



I. D«F. 23] NOTE ON DEFINITION 23 I93 

We will now take the three groups in order. 

(i) The first observation of Schotten is that the varieties of this group 
which regard parallels as (a) meeting at infinity or (d) having a common 
point at mfinity (first mentioned apparentlv by Kepler, 1604, as a '*fa^on de 
parler" and then used by Desaigues, 1639) are at least unsuitable definitions 
tor elementary text-books. How do we know that the lines cut or meet at 
infinity? We are not entitled to assume either that they do or that they do 
not, because "infinity'' is outside our field of observation and we cannot verify 
either. As Gauss says (letter to Schumacher), " Finite man cannot claim to 
be able to regard the infinite as something to be grasped by means of ordinary 
methods of observation." Steiner, in speaking of the rays passing through a 
point and successive points of a straight line, observes that as the point of 
intersection gets further away the ray moves continually in one and the same 
direction ("nach einer und derselben Richtung hin"); only in one position, 
that in which it is parallel to the straight line, "there is no real cutting^* 
between the ray and the straight line ; what we have to say is that the ray is 
*^ directed towards the infinitely distant paint an the straight line.^^ It is true 
that higher geometry has to assume that the lines do meet at infinity: whether 
such lines exist in nature or not does not matter (just as we deal with "straight 
lines " although there is no such thing as a straight Une). But if two lines do 
not cut at any finite distance, may not the same thing be true at infinity also ? 
Are lines conceivable which would not cut even at infinity but always remain 
at the same distance from one another even there ? Take the case of a line 
of railway. Must the two rails meet at infinity so that a train could not stand 
on them there (whether we could see it or not makes no difference)? It 
seems best therefore to leave to higher geometry the conception of infinitely 
distant points on a line and of two straight lines meeting at infinity, like 
imaginary points of intersection, and, for the purposes of elementaiy geometry, 
to rely on the plain distinction between "parallel" and "cutting" which 
average human mtelligence can readily grasp. This is the method adopted 
by Euclid in his definition, which of course belongs to the group (i) of 
definitions regarding parallels as non-secant 

It is significant, I think, that such authorities as Ingrami {Elementi di 
geometria^ 1904) and Enriques and Amaldi (Elementi di geometria^ 1905)1 
after all the discussion of principles that has taken place of late years, give 
definitions of parallels equivalent to Euclid's : " those straight lines in a plane 
which have not any point in common are called parallels." Hilbert adopts 
the same point of view. Veronese, it is true, takes a different line. In his 
great work Fondamenti di geametria, 1 891, he had taken a ray to be parallel to 
another when a point at infinity on the second is situated on the first ; but he 
appears to have come to the conclusion that this definition was unsuitable for 
his Elementi. He avoids however giving the Euclidean definition of parallels 
as "straight lines in a plane which, though produced indefinitely, never meet," 
because "no one has ever seen two straight lines of this sort," and because 
the postulate generally used in connexion vnth this definition is not evident in 
the way that, m the field of our experience, it is evident that only one straight 
line can pass through two points. Hence he gives a different definition, for 
which he claims the advantage that it is independent of the plane. It is 
based on a definition of figures " opposite to one another with respect to a 
point" (or reflex figures). "Two figures are opposite to one another with 
respect to a point C7, e.g. the figures ABC ... and A'ffC.,.^ M to every point 
of the one there corresponds one sole point of the other, and if the segments 

H. s. 13 



t94 6001C t [i. DBF. tj 

OA^ OB^ OCy ... joining the points of one figure to C7 are respectively equal 
and opposite to the segments 0A\ OB^ 0C\ ... joining to O the corresponding 
points of the second " : then, a iramsverstU of two straight lines being any 
segment having as its extremities one point of one line and one point of the 
other, " two straight lines are aUlei parallel if one of them contains two foinis 
opposite to two points of the other with reject to the middle point of a common 
transverscU,^^ It is true, as Veronese says, that the paiallek so defined and the 
parallels of Euclid are in substance the same; but it can hardly be said diat 
the definition gives as good an idea of the essential nature of parallels as does 
Euclid's. Veronese has Xjoprove^ of course, that his parallels have no point in 
common, and his ''Postulate of Parallds" can hardly be called more evident 
than Euclid's : " If two straight lines are parallel, they are figures opposite to 
one another with respect to the middle points of all their transversal segments." 

(2) The direction-xheoiy. 

The fallacy of this theory has nowhere been more completely exposed 
than by C. L. Dodgson (EuJid and his modem Moals, 1879). According to 
Killing {Einfuhrung in die Grundlagm der Geometrie^ i. p. 5) it would appear 
to have originated with no less a person than Leibniz. In the text-books 
which employ this method the notion of direetion appears to be regarckd as a 
primary, not a derivative notion, since no definition is given. But we ought 
at least to know how the same direction or like directions can be recognised 
when two different straight lines are in question. But no answer to this 
question is forthcoming. The fact is that the whole idea as applied to non- 
coincident straight lines is derived firom knowledge of the properties of 
parallels ; it is a case of explaining a thing by itself. The idea of parallels 
being in the same direction perhaps arose from the conception of an angle as 
a differefue of direction (the hoUowness of which has already been expmed) ; 
sameness of direction for parallels follows from the same "difference of 
direction" which both exhibit relatively to a third line. But this is not 
enough. As Gauss said {Werhe, iv. p. 365), " If it [identity of direction] is 
recognised by the equality of the angles formed with one third straight bne, 
we do not yet know vnthout an antecedent proof whether this same equality 
will also be found in the angles formed with a fourth straight line " (and any 
number of other transversals) ; and in order to make this theory of parallels 
valid, so far from getting rid of axioms such as Euclid's, you would have to 
assume as an axiom what is much less axiomatic, namely that "straight lines 
which make equal corresponding angles vnth a certain transversal do so with 
any transversal" (Dodgson, p. loi). 

(3) In modem times the conception of parallels as equidistant straight 
lines was practically adopted by Clavius (the editor of Euclid, bom at 
Bamberg, 1537) and (according to Saccheri) by Borelli {Euclides restitutus^ 
1658) although they do not seem to have defined parallels in this way. 
Saccheri points out that, before such a definition can be used, it has to 
be proved that " the geometrical locus of points equidistant fi^m a straight J 
line is a straight line." To do him justice, Clavius saw this and tried to 
prove it: he makes out that the locus is a straight line according to the 
definition of Euclid, because "it lies evenly with respect to all tiie points 
on it"; but there is a confusion here, because such "evenness" as the locus 
has is with respect to the straight line fi^m which its points are equidistant, 
and there is noUiing to show that it possesses this property with respect 
to itself. In fact the theorem cannot be proved vrithout a postulate. 



J 



I I. Post, i] N0T£ ON t>OStULATfi i tpj 

Postulate i. 

HinioBia dwo toitos arjfjLtlov hci xav (rqfUiov cMciav ypafjLfirjy ayayctv. 
Let the following be postulated: to draw a straight line from any point to 
any point. 

From any point to any point. In general statements of this kind 
the Greeks did not say, as we do, "a/iy point," ^*any triangle" etc., but 
^* every point," ^^ every triangle" and the like. Thus the words are here 
literally "from every point to every point." Similarly the first words of 
Postulate 3 are " with every centre and distance," and the enunciation, e.g., of 
I. 1 8 is " In every triangle the greater side subtends the greater angle." 

It will be remembered that, according to Aristotle, the geometer must in 
general assume what a thing is, or its definition, but must prove that it is, 
i.e. the existence of the thing corresponding to the definition : only in the case 
of the two most primary things, points and lines, does he assume, without 
proof, both the definition and the existence of the thing defined. Euclid has 
mdeed no separate assumption affirming the existence oi points such as we find 
nowadays in text-books like those of Veronese, Ingrami, Enriques, "there exist 
distinct points" or "there exist an infinite number of points." But, as re- 
gards the only lines dealt with in the Elements^ straight lines and circles, 
existence is asserted in Postulates i and 3 respectively. Postulate i however 
does much more than (i) postulate the existence of straight lines. It is 
(2) an answer to a possible objector who should say that you cannot, with the 
imperfect instruments at your disposal, draw a mathematical straight line at all, 
and consequently (in the words of Aristotle, Anal, post. i. 10, 76 b 41) that 
the geometer uses ^se hypotheses, since he calls a line a foot long when it is 
not or straight when it is not straight. It would seem (if Gherard's translation 
is right) that an-Nairizi saw that one purpose of the Postulate was to refute 
this criticism : " the utility of the first three postulates is (to ensure) that the 
weakness of our equipment shall not prevent (scientific) demonstration" 
(ed. Curtze, p. 30). The fact is, as Aristotle says, that the geometer's demon- 
stration is not concerned with the particular imperfect straight line which he 
has drawn, but with the ideal straight line of which it is the imperfect 
represer^tation. Simplicius too indicates that the object of the Postulate is 
rather to enable the drawing of a mathematical straight line to be imagined 
than to assert that it can actually be realised in practice : " he would be a 
rash person who, taking things as they actually are, should postulate the 
drawing of a straight line from Aries to Libra." 

There is still something more that must be inferred from the Postulate 
combined with the definition of a straight line, namely (3) that the straight 
line joining two points is unique : in other words that, tf tivo straight lines 
("rectilineal segments," as Veronese would call them) luwe the same extremities^ - 
they must coinade throughout their length. The omission of Euclid to state 
this in so many words, though he assumes it in i. 4, is no doubt answerable for 
the interpolation in the text of the equivalent assumption that two straight 
lines cannot enclose a space^ which has constantly appeared in mss. and editions 
of Euclid, either among Axioms or Postulates. That Postulate i included it, 
by conscious implication, is even clear from Proclus* words in his note on l 4 
(p. 239, 16) : "therefore two straight lines do not enclose a space, and it was 
with knowledge of this fact that the writer of the Elements said in the first of 
his Postulates, to draw a straight line from any point to any pointy implying 
that it is ^M^ straight line which would always join the two points, not two.^ 

13— » 



I 



19^ 



BOOK I 



[l Post. 



i»s 




Proclus attempts in the same note (p. 339) to P^yoe that two straight lines 
cannot enclose a space, using as his bws the definition of the diameter of a 
circle and the theorem, stat^ in it, that any diameter divides tfie cirde into 
two equal parts. ' 

Suppose, he says, ACB^ ADB to be two straight lines enclosing a q»oe. 
Produce them (beyond B) indefinitely. With centre B 
and distance AB describe a circle, cutting the lines so 
produced va F^ E respectively. 

Then, since ACBF, ADBE are both diameters 
cutting off semi-circles, the arcs AE^ AEF are equal : 
which is impossible. Therefore etc. 

It ¥nll be observed, however, that the straight lines 
produced are assumed to meet the circle given in two 
different points E^ F^ whereas, for anything we know, 
Ey F might coincide and the straight lines have three common points. The 
proof is therefore delusive. 

Saccheri gives a different {xoof. From Euclid's definition of a straight 
line as that which lies evenly with its points he infers that, when 
such a line is turned about its two extremities, which remain fixed, 
all the points on it must remain throughout in the same position, and 
cannot take up different positions as the revolution proceeds. " In 
this view of the straight line the truth of the assertion that two 
straight lines do not enclose a space is obviously involved. In fact, 
if two lines are given which enclose a s^ce, and of which the two 
points A and X are the common extremities, it is easily shown that 
neither, or else only one, of the two lines is straight" 

It is however better to assume as a postuiate the fact, inseparably 
connected with the idea of a straight line, that there exists onfy one straight 
line containing two given points^ or, if two straight lines have two points in 
commotiy they coiticide throughout. 

Postulate 2. 

Kal icaetpaxT\kivyfv tMtiav Kara to (twc^^ i'f* cMctac ^ic/SaXciy. 

To produce a finite straight line continuously in a straight line, 

I translate icvtrmaxr\tivrjpf by finite^ because that is the received equivalent, 
and because any alternative word such as limited^ terminated^ if applied to a 
straight line, would equally fail to express what modem Italian geon^eters aptly 
call a rectilineal segment^ that is, a straip;ht line having two extremities. 

Just as Post. I asserting the possibility of drawing a straight line from any 
one point to another must be held to declare at the same time that the 
straight line so drawn is unique, so Post 2 maintaining the possibility of 
producing a finite straight line (a "rectilineal segment ") continuously m a 
straight hne must also be held to assert that the straight line can only be 
produced in one way at either end, or that the produced part in either 
direction is unique ; in other words, that two straight lines cannot have a 
common segment. This latter assumption is not expressly appealed to by 
Euclid until xi. i. But it is needed at the very beginning of Book'i. Proclus 
(p. 314, 18) says that 2^no of Sidon, an Epicurean, maintained that the very 
first proposition i. i requires it to be admitted that *' two straight lines cannot 
have the same segments " ; otherwise AQ BC might meet before they arrive 
at C and have the rest of their length common, in which case the actual 
triangle formed by them and AB would not be equilateral. The assumption 
that two straight lines cannot have a common segment is certainly necessary 
in I. 4, where one side of one triangle is placed on that side of the other 



( 



j L Post. 2] 



NOTES ON POSTULATES i. 2 



197 




triangle which is equal to it, and it is inferred that the two coincide throughout 

their length : this would by no means follow if two straight lines could have a 

common segment Proclus (p. 215, 24), while observing that Post. 2 clearly 

indicates that the produced portion must be one^ attempts to prove it, but 

unsuccessfully. Both he and Simplicius practically 

use the same argument. Suppose, says Proclus, 

that the straight lines ACy AD have AB as a 

common segment. With centre B and radius BA 

describe a circle (Post 3) meeting AC^ AD in 

C, D. Then, since ABC is a straight line through 

the centre, AEC is a semi-circle. Similarly, ABD 

being a straight line through the centre, AED is a 

semi-circle. Therefore AEC is equal to AED\ 

which is impossible. 

Proclus observes that 2^no would object to this proof as really depending 
on the assumption that "two circumferences (of circles) cannot have one 
portion common " ; for this, he would say, is assumed in the common proof 
by superposition of the fact that a circle is bisected by a diameter, since that 
proof takes it for granted that, if one part of the circumference cut off by the 
diameter, when applied to the other, does not coincide with it, it must neces- 
sarily fall either entirely outside or entirely inside it, whereas there is nothing 
to prevent their coinciding, not altogether, but in part only ; and, until you 
really prove the bisection of a circle by its diameter, the above proof is not 
valid. ' Posidonius is represented as having derided Zeno for not seeing that 
the proof of the bisection of a circle by its diameter goes on just as well if the 
circumferences fail to coincide in part only. But the true objection to the 
proof above given is that the proof of the bisection of a circle by any diameter 
i/f^'^ assumes that two straight lines cannot have a common segment; for, if 
we wish to draw the diameter of a circle which has its extremity at a given point 
of the circumference we have to join the latter point to the centre (Post i) and 
then to produce the straight line so drawn till it meets the circle again (Post 2), 
and it is necessary for the proof that the produced part shall be unique. 

Saccheri adopted the proper order when he gave, first the proposition that 
two straight lines cannot have a common s^ment, and after that the 
proposition that any diameter of a circle bisects the circle and its circumference. 

Saccheri's proof of the former is very interesting as showing; the thorough- 
ness of his method, if not at the end entirely convincing. It is in five stages 
which I shall indicate shortly, giving the full argument of the first only. 

Suppose, if possible, that AX is a common segment of both the straight 
lines AXB^ AXC^ in one plane, produced beyond 
X. Then describe about X as centre, with radius 
XB or XC^ the arc BMC^ and draw through X to 
any point on it the straight line XM. 

(1) I maintain that, with the assumption 
made, the line AXM is also a straight line which 
is drawn from the point A to the point X and pro- 
duced beyond X. 

For, if this line were not straight, we could draw 
another straight line AM which for its part would 
be straight This straight line will either (a) cut one 
of the two straight lines XB^ XC in a certain point 
K or (b) enclose one of them, for instance XBy in 
the area bounded by AX^ XMsind APLM. 




193 BOOK I [lPost. i 

But the first alternative (a) obviously contndicU the forq^oiiig lemma rtliat 
two straight lines cannot enclose a space^ since in that case the two lines 
AXJ^, ATK^ which by hypothesis are straight, would enclose a space. 

The second possibility ijf) is at once seen to invdve a similar abaurdity. 
For the straight line XB must, when produced beyond B^ ultimatel]r meet 
APLM in a point Z. Consequently the two lines AXBL^ APL^ which by 
hypothesis are straight, would again enclose a space. If however we were to 
assume that the straight line XB produced beyond B will ultimately meet 
either the straight line XMor the straight line XA in another point, we should 
in the same way arrive at a contradiction. 

From this it obviously follows that, on the assumption made, the line 
AXM\% itself the straight line which was drawn from the point ^ to the point 
M'y and that is what was maintained. 

The remaining stages are in substance these. 

(ii) Iftke straight line AXB, regarded as rigid^ revohes about AX as axis^ 
U cannot assume two more positions in the same fiane^ so tkat^ for exam^e^ in 
one position XB should coincide with XC, and in the other with XM. ^ 

[This is proved by considerations of symmetry. AXB cannot be altogether 
** similar or equal to " AXC^ if viewed from the same side (left or ri^t) of 
both : otherwise they would coincide, which by hypothesis they do not But 
there is nothing to prevent AXB viewed from one side (sav the left) being 
'* similar or equal to '' AXC viewed from the other side (i.e. the right), so that 
AXB can^ without any chanffe, be brought into the position AXC. 

AXB cannot however tale the position of the other straight line AXAitm i 
well If they were like on one side, they would coincide; if they were like on 1 
opposite sides, AXM^ AXC would be like on the same side and therefore j 
comdde.] I 

(iii) The other positions of AXB during the revolution must be above or 4 
below the original plane. 4 

(iv) It is next maintained that there is a point l^ on the arc BC such that^ if 1 
XD is drffwn^ AXD is not only a straight line but is such that viewed from the left I 
side it is exactly ^^ similar or equaV^ to what it is when viavedfrom the right side, > 

{Firsts it is proved that points M^ Fcan be found on the arc, corresponding \ 
in the same way as B^ C do, but nearer together, and of course AXM^ AXF 
are both straight lines. 

Secondly^ similar corresponding points can be found still nearer together, 
and so on continually, until either (a) we come to one point D such that AXD \ 
is exactly iihe itself when the right and left sides are compared^ or (b) there are ; 
two ultimate points of this sort M^ F^ so that both AXM^ AXF have this 
property. 

Thirdly^ (b) is ruled out by reference to the definition of a straight line. ' 

Hence (a) only is true, and there is only one point D such as described.] 

(v) Lastly, Saccheri concludes that the straight line AXD so determined i 
*' is alone a straight line, and the immediate prolongation from A beyond X to 
/>," relying again on the definition of a straight line as 'Mying evenly." 

Simson deduced the proposition that two straight lines cannot have a 
common segment as a corollary from i. 1 1 ; but his argument is a complete 
petitio principii^ as shown by Todhunter in his note on lliat proposition. 

Produs (p. SI 7, lo) records an ancient proof also based on the proposition 
I. 1 1. Zeno, he says, propounded this proof and then critidsed it 




I. Post- 2, 3] NOTES ON POSTULATES 2, 3 199 

Suppose that two straight lines AC^ AD have a common segment AB^ and 
let BE be drawn at right angles to AC. 

Then the angle BBC is right. 

If then the angle EBD is also right, the two 
angles will be equal : which is impossible. 

If the angle EBD is not right, draw ^^at right 
angles to AD \ therefore the angle FBA is right. 

But the angle EBA is right 

Therefore the angles EBA^ FBA are equal : 
which is impossible. 

Zeno objected to this, says Proclus, because it assumed the later pro- 
position I. 1 1 for its proof. Posidonius said that there was no trace of such 
a proof to be found in the text-books of Elements, and that it was only invented 
by Zeno for the purpose of slandering contemporary geometers. Posidonius 
maintains further that even this proof has something to be said for it. There 
must be some straight line at right angles to each of the two straight lines A C, 
AD (the very definition of right angles assumes this): ^^ suppose /Aen U happens 
to be the straight line we have set upJ* Here then we have an ancient instance 
of a defence of hypothetical construction^ but in such apologetic terms (" it is 
possible to say something even for this proof") that we may conclude that in 
general it would not have been accepted by geometers of that time as a 
Intimate means of proving a proposition. 

Todhunter proposed to deduce that two straight lines cannot have a 
common segment from i. 13. But this will not serve either, since, as before 
mentioned, the assumption is really required for i. 4. 

It is best to make it a postulate. 

Postulate 3. 

Kal Tovrl iccFrpcp jcal Zuurrrnijari kvkXjO¥ ypa^cotfai. 

To describe a circle with any centre and distance. 

In this case Euclid's Jext has the passive of the verb: '*a circle can be 
drawn " ; Proclus however has the active (ypJ^oi) as Euclid has in the first 
two Postulates. 

Distance^ iuKm/fiaTL This word, meaning ** distance *' quite generally (cf. 
Arist Metaph. 1055 a 9 *'it is between extremities that distance is greatest," 
ibid. 1056 a 36 '' things which have something between them, that is, a certain 
distance "), and also " distance " in the sense of '' dimension " (as in " space 
has three dimensions, length, breadth and depth," Arist. Physics iv. i, 209 a 4), 
was the regular word used for describing a circle with a certain radius^ the 
idea being that each point of the circumference was at that distance from the 
centre (cf. Arist. Meteorologica in. 5, 376 b 8 : "if a circle be drawn. ..with 
distance Mil "). The Greeks had no word corresponding to radius : if they 
had to express it, they said "(straight lines) drawn from the centre" (al Ik rov 
Kivrpov, Eucl. III. Def. I and Prop. 26; Meteorologica 11. 5, 362 b i has the full 
phrase ol Ik tov mtyrpov dyofuyiu ypofifjuoi), 

Mr Frankland observes that it would be remarkable if, unlike Postulates i 
and 2, this Postulate implied merely what it says, that a circle can be drawn 
with any centre and distance. We may r^ard it, if we please, as helping to the 
complete delineation of the Space which Euclid's geometry is to investigate 
formally. The Postulate has the effect of removing any restriction upon the 
size of the circle. It may (i) be indefinitely small, and this implies that space 
is continuous, not discrete, with an irreducible minimum distance between 



soo BOOK I [l Post 3, 4 

contiguous pomts in it (a) The drde may be indefinitely bu8e» wfaidi 
implies the fundamental hypothecs of h^miudi of space. This last assumed 
characteristic of space is essential to the proof of i. i6^ a theorem not 
universally valid in a space whidi is unbounded in extent but finite in sise. It 
would however be unsafe to suppose that Euclid foresaw the use to which his 
Postulate might thus be put, or formulated it with such an intention. 

Postulate 4. 

Koi irauiif ths 6p$a9 yta^tom Itrat oXXipXaif Aoi. 

ITiat all right angles are equal to 4me another. 

While this Postulate asserts the essential truth that a risht angle is a 
determinate magnitude so that it really serves as an invariable standard by 
which other (acute and obtuse) angles may be measured, much more than 
this is implied, as will easily be seen firom the following consideratioiL ^ If the 
statement is to be proved^ it can only be proved b^ the method of applying one 
pair of right angles to another and so arguing their equality. But this method 
would not be valid unless on the assumption of the invaruMlity ofjigiirest 
which would therefore have to be asserted as an antecedent postulate. Euclid 
preferred to assert as a postulate, directly, the fact that all right angles are 
equal; and hence his postulate must be tidten as equivalent to me principle of 
invariability of figures or its equivalent, the homogeneity of space. 

According to Proclus, Geminus held that this Postulate should not be 
classed as a postulate but as an axiom, since it does not, like the first three 
Postulates, assert the possibility of some amstntction but expresses an essential 
property of right angles. Produs further observes (p. 188, 8) that it is not a 
postulate in Aristotle's sense either. (In this I think he is wrong, as exjdained 
above.) Proclus himself, while regarding the assumption as axiomatic (** the 
equality of right angles suggests itself even by virtue of our common notions"), 
is prepared with a proof, if such is asked for. 

Let ABC, DEF be two right 
angles. 

If they are not equal, one of them 
must be the greater, say ABC. 

Then, if we apply DE to AB, EF H- 
will fall within ABC, as BG. 

Produce CB to H. Then, since 
ABC is a right angle, so is ABH, and the two angles are equal (a right angle 
being by definition equal to its adjacent angle). 

Therefore the angle ABH\% greater tlum the angle ABG. 

Producing GB to K, we have similarly the two angles ABK, ABG both 
right and equal to one another; whence the angle ABHS& less than the angle 
ABG. 

But it is also greater : which is impossible. 

Therefore etc. 

A defect in this proof is the assumption that CB, GB can each be 
produced only in one way, and that ^9^ falls outside the angle ABH. 

Saccheri's proof is more careful in that he premises a third lemma in 
addition to those asserting (i) that two straight lines 
cannot enclose a space and (2) that two straight lines 
cannot have a common segment The third lemma is : 
If two straight Unes AB, CXD meet one another at an 
intermediate point X, they do not touch at that point, but 
cut one another. 




1 

i 

I 



1. Post. 4] 



NOTES ON POSTULATES 3, 4 



aoi 



M 



Suppose now that DA standing on BAC makes the two angles DAB^ 
Z>^C equal, so that each is a right angle by the definition. 

Similarly, let LHioxm with the straight line FHM the right angles LHFy 
LHM. 

Let DA^' HL be equal ; and sup- 
pose the whole of the second figure 
so laid upon the first that the point 
^falls on A^ and L on Z>. 

Then the straight line FHMmW 
(by the third lemma) not touch the 
straight line BC at A ; it will either 

(a) coincide exactly with BC, or 

(d) cut it so that one of its extremities, as F, will fall above [BC] and the 
other, M^ below it. 

If the alternative (a) is true, we have already proved the exact equality of 
all rectilineal right angles. 

Under alternative (b) we prove that the angle LHF, being equal to the 
angle DAF, is less than the angle DAB or DAC, and a fortiori less than the 
angle DAM ox LHM\ which is contrary to the hypothesis. 

[Hence (a) is the only possible alternative, so that all right angles are 
equal.] 

Saccheri adds that it makes no difference if the angle DAF diverges 
infinitely little from the angle DAB, This would equally lead to a conclusion 
contradicting the hypothesis. 

It will be observed that Saccheri speaks of ''the exact equality of all 
rectilineal right angles." He may have had in mind the remark of Pappus, 
quoted by Proclus (p. 189, 11), that the converse of 
this postulate, namely that an angle which is equal 
to a right angle is also right, is not necessarily true, 
unless the former angle is rectilinecU. Suppose two 
equal straight lines BA^ BC9X right angles to one 
another, and semi-circles described on BA, BC 
respectively as AEB, BDC in the figure. Then, 
since the semi-circles are equal, tiiey coincide if 
applied to one another. Hence the ''angles ** 
EBA, DBC are equal Add to each the " angle " 
ABD ; and it follows that the lunular angle EBD is equal to the right angle 
ABC. (Similarly, if BA, BC be inclined at an acute or obtuse angle, instead 
of at a rfght angle, we find a lunular angle equal to an acute or obtuse angle.) 
This is one of the curiosities which Greek commentators delighted in. 

Veronese, Ingrami, and Enriques and Amaldi deduce the fact that all 
right angles are equal from the equivalent fact that all fiat angles are equals 
which is either itself assumed as a postulate or immediately deduced from some 
other postulate. 

HUbert takes quite a different line. He considers that Euclid did wrong 
in placing Post 4 among "axioms." He himself, after his Group in. of 
Axioms containing six relating to congruence, proves several theorems about 
the congruence of triangles and angles, and then deduces our Postulate. 

As to the raison iPitre and the place of Post 4 one thing is quite certain. 
It was essential from Euclid's point of view that it should come before Post. 5, 
since the condition in the latter that a certain pair of angles are together less 
than two right angles would be useless unless it were first made clear Ihat 
right angles are angles of determinate and invariable magnitude. 




ao3 BOOK I [i. Post. 5 1 

Postulate 5. 

Kcu iay etc Svo c^ctat cMcia i§iwiwr€vaa rkt hrri^ mu hn rk aira fU/ni vtMit 
Suo 6ft6mv iXaaaoya^ iroc^ iKPaXXofiiva% r&« Svo cMctat Jv* avcipor ov/cviVTCcr, 

7>(a/, (/*« straight line falling on two straighl lines make ike interior angles 
on the same side less than two right angles^ the two straight Knes^ if prodneed 
indefinitely^ meet on that side on which an the angles less than the two rfght 
angles. 

Although Aristotle gives a dear idea of what he understood by a postulaie^ 
he does not give any instances from g^metry; still less has he an^ allusion 
recalling the particular postulates found in Euclid. We naturally infer that 
the formulation of these postulates was Euclid's own work. There is a more 
positive indication of the originality of Postulate 5* since in the passage {Anal, 
prior. II. 16, 65 a 4) quoted above in the note on the definition of parallels he 
alludes to some petitio primipii involved in the theory of parallels current in 
his time. This reproach was removed by Euclid when he laid down this 
epoch-making Postulate. When we ocmsider the countless successive attempt 
made through more than twenty centuries to prove the Postulate, many of 
them by geometers of abili^, we cannot but admire the genius of the man 
who conduded that such a hypothesisi which 'he found necessary to the 
validity of his whole system of geometiy, wasjc^ly indem(mstmble. 

From the very b^mning» as we know irom ProcIus| the Tostulate was 
attacked as such, and attempts were made to prove it as a theorem or to get 
rid of it by adopting some odier definition of jparallds; while in modem times 
the literature of the subject is enormous. Riccardi {^aggju^ di una bibU^^fta 
Euclidea^ Part iv., Bologna, 1890) has twenty quarto pages of titles of mono- 
graphs relating to Post 5 between the dates 1607 and 1887. Max Simon 
\Ueber die Entwichlung der Elementar-geometrie im XIX. Jahrhundtrt^ 1906) 
notes that he has seen three new attempts, as late as 1891 (a century after 
Gauss laid the foundation of non-Euclidean geometry), to prove the theory of 
' parallels independently of the Postulate. Max Simon himself (pp. 53 — 61) 
gives a large number of references to books or articles on the subject and 
refers to the copious information, as to contents as wdl as names, con- 
tained in Schotten's Inhalt und Methodt des planimetrischen Unterrichts^ IL 

PP- 1*3— 33»- 

This note will include some account of or allusion to a few of the most 
noteworthy attempts to prove the Postulate. Only those of andent times, as 
being less generally accessible, will be described at any length; shorter 
references must suffice in the case of the modem geometers who have made 
the most important contributions to the discussion of the Postulate and have 
thereby, in particular, contributed most towards the foundation of the non- 
Euclidean geometries, and here I shall make use prindpally of the valuable 
Article 6, Sulla teoria delle parallele e sulk geomeirie nem-euclidee (by Roberto 
Bonola), in Ouestioni riguardanti la geametria elementare (pp. 143 — 222). 

Proclus (p. 191, 21 sqq.) states very clearly the nature of the first 
objections taken to the Postulate. 

*'This ought even to be struck out of the Postulates altogether; for it is a 
theorem involving many difficulties, which Ptolemy, in a certain book, set 
himsdf to solve, and it requires for the demonstration of it a number 
of definitions as well as the(»ems. And the converse of it is actually 
pioved by Eudid himsdf as a theorem. It may be that some would be 



I 



I. Post. 5] NOTE ON POSTULATE 5 203 



1 deceived and would think it proper to place even the assumption in question 
among the postulates as affording, in the lessening of the two right angles, 
ground for an instantaneous belief that the straight lines converge and meet. 
To such as these Geminus correctly replied that we have learned from the 
very pioneers of this science not to have any regard to mere plausible imagin- 
ings when it is a question of the reasonings to be included in our geometrical 
doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of 
a rhetorician as to accept mere plausibilities from a geometer; and Simmias is 
made by Plato to say that he recognises as quacks those who fashion for 
themselves proofs from probabilities. So in this case the fact that, when the 
right angles are lessened, the straight lines converge is true and necessary; 
but the statement that, since they converge more and more as they are pro- 
duced, they will sometime meet is plausible but not necessary, in the absence 
of some argument showing that this is true in the case of straight lines. For 
the fact that some lines exist which approach indefinitely, but yet remain 
non-secant (dav/LnrrcDroi), although it seems improbable and paradoxical, is 
nevertheless true and fully ascertained with regard to other species of lines. 
May not then the same thing be possible in the case of straight lines which 
happens in the case of the lines referred to ? Indeed, until the statement in 
the Postulate is clinched by proof, the facts shown in the case of other lines 
may direct our imagination the opposite way. And, though the controversial 
arguments against the meeting of the straight lines should contain much that 
is surprising, is there not all the more reason why we should expel from our 
body of doctrine this merely plausible and unreasoned (hypothesis) ? 

"It is then clear from this that we must seek a proof of the present 
theorem, and that it is alien to the special character of postulates. But how 
it should be proved, and by what sort of arguments the objections taken to 
it should be removed, we must explain at the point where the writer of the 
Elements is actually about to recall it and use it as obvious. It vnll be 
necessary at that stage to show that its obvious character does not appear 
independendy of proof, -but is turned by proof into matter of knowledge." 

Before passing to the attempts of Ptolemy and Proclus to prove the 
Postulate, I should note here that Simplicius says (in an-NairIzi, ed. Besthom- 
Heiberg, p. 119, ed. Curtze, p. 65) that this Postulate is by no means manifest, 
but requires proof, and accordingly '* Abthiniathus " and Diodorus had 
already proved it by means of many different propositions, while Ptolemy also 
had explained and proved it, using for the piffpose Eucl. i. 13, 15 and 16 (or 
18). The Diodorus here mentioned may be the author of the Analemma on 
which Pappus wrote a commentary. It is difficult even to frame a conjecture 
as to who " Abthiniathus " is. In one place in the Arabic text the name 
appears to be written " Anthisathus " (H. Suter in Zeitschrift fiir Math, und 
Pkysiky xxxviii.y hist, litt Abth. p. 194). It has occurred to me whether he 
might be Peithon, a friend of Serenus of Antinoeia (Antinoupolis) who was 
long known as Serenus of Antissa, Serenus says (De sectione cylindri^ ed. 
Heiberg, p. 96): "Peithon the geometer, explaining parallels in a work of his, 
was not satisfied with what Euclid said, but showed their nature more cleverly 
by an example; for he says that parallel straight lines are such a thing as we 
see on walls or on the ground in the shadows of pillars which are made when 
either a torch or a lamp is burning behind them. And, althougn this has only 
been matter of merriment to every one, I at least must not deride it, for the 
respect I have for the author, who is my friend.'' If Peithon was known as 
" of Antinoeia " or " of Antissa,** the two forms of the mysterious name might 
perhaps be an attempt at an equivalent; but this is no more than a guess. 



304 BOOK I [i. Post. 5 

Simplicius adds in full and word for word the attempt of his "friend** or 
his *' master Aganis " to prove the Postulate. 

Proclus returns to the subject (p. 365, 5) in his note on EucL i. 39. He 
says that before his time a certain numb^ of g^meters had classed as a 
theorem this Euclidean postulate and thought it matter for proof, and he then 
proceeds to give an account of Ptolem/s argument 

Noteworthy attempts to prove the Postulate. 

Ptolemy. 

We learn from Proclus (p. 365, 7 — 11) that Ptolemy wrote a book on the 
proposition that " straight lines drawn from angles less than two right angles 
meet if produced," and that he used in his ''proof" many of the theorems in 
Euclid preceding i. 29. Proclus excuses himself from reproducing the nrly 
part of Ptolemy's argument, only mentioning as one of the propositions 
proved in it the theorem of EucL i. a8 that, if two straight lines meeting a 
transversal make the two interior angles on the same side equal to two ri^t 
angles, the straight lines do not meet, however far produced. 

I. From Proclus' note on 1. 28 (p. 362, 14 sq.) we know that Ptdemy 
proved this somewhat as follows. 

Suppose that there are two straight lines AB, CD, and that EFGH, 
meeting them, makes the angles BFG, FGD equal to two right angles. 
I say that AB, CD are paralH that is, they 
are non-secant 

For, if possible, let FB, GD meet at K. 

Now, since the angles BFG, FGD are 
equal to two right angles, while the four 
angles AFG, BFG, FGD, FGC are together 
equal to four right angles, 

the angles AFG, FGC are equal to two 
right angles. 

^^If therefore FB, GD, when the interior angles are equal to two right 
angles, meet at K, the straight lines FA, GC will also meet if produced; for the 
angles AFG, CGFaie also equal to two right angles. 

''Therefore the straight lines will either meet in both directions or in 
neither direction, if the two pairs of interior angles are both equal to two right 
angles. 

" Let, then, FA, GC meet at Z. 

"Therefore the straight lines LABK^ LCDK enclose a space : which is 
impossible. 

"Therefore it is not possible for two straight lines to meet when the 
interior angles are e^ual to two right aisles. Therefore they are parallel" 

[The argument m the words italic^ed would be clearer if it had been 
shown that the two interior angles on one side of EH are severally equal to the 
two interior angles on the other, namely BFG to CGF and FGD to AFG\ 
whence, assuming FB, GD to meet in K, we can take the triangle KFG and 
place it (e.g. by rotating it in the plane about O the middle point of FG) so 
diat FG falls where GF\& in the figure and GD fdls on FA, in which case 
FB must also frdl on GC\ hence, since FB, GD meet at K, GC and FA 
must meet at a corresponding point Z. Or, as Mr Frankland does, we may 
substitute for FG a straight line MN through O the middle point of FG 
drawn peipendicular to one of the parallels, say AB, Then, smce the two 
triangles OMF, ONG have two angles equal respectively, namely FOM to 




r 

z 



1. Post. 5] NOTE ON POSTULATE 5 . 205 

GON(i. is) and OFMio OGN, and one side O-^ equal to one side OG^ the 
triangles are congruent, the angle ONG is a right angle, and MN is perpen- 
dicular to both AB and CD. Then, by the same method of application, 
MA^ NC are shown to form with MN di triangle MALCN congruent with 
the triangle NDKBM, and MA^ NC meet at a point L corresponding to K. 
Thus the two straight lines would meet at the two points K^ JL This is what 
happens under the Riemann hypothesis, where the axiom that two straight 
lines cannot enclose a space does not hold, but all straight lines meeting in 
one point have another point common also, and e.g. in the particular figure 
just used K^ L are pomts common to all perpendiculars to MN If we 
suppose that K^ L are not distinct points, but om point, the axiom that two 
straight lines cannot enclose a space is not contradicted.] 

II. Ptolemy now tries to prove i. 29 without using our Postulate, and 
tl.wii deduces the Postulate from it (Proclus, pp. 365, 14 — 367, 27). 

The argument to prove i. 29 is as follows. 

The straight line which cuts the parallels must make the sum of the 
interior angles on the same side equal to, greater 
than, or less than, two right angles. ^ ^ ? 

"Let AB, CD be parallel, and let FG meet 
them. I say (i) that FG does not make the 
interior angles on the same side greater than two g — 
right angles. 

" For, if the angles AFG, CGF are greater than two right angles, the 
remaining angles BFG, DGF are less than two right angles. 

*' But the same two angles are also greater than two right angles ; for AF, 
CG are no more parallel than FB, GD, so that, if the straight line falling on 
AF, CG makes the interior angles greater than two right angles, the straight line 
falling on FB, GD will also make the interior angles greater than two right 
angles. 
V " But the same angles are also less than two right angles ; for the four 
angles AFG, CGF, BFG, DGF are equal to four right angles : 
which is impossible. 

"Similarly (2) we can show that the straight line falling on the parallels 
does not make the interior angles on the same side less than two right angles. 

" But (3), if it makes them neither greater nor less than two right angles, 
it can only make the interior angles on the same side equal to two right 
angles.*' 

III. Ptolemy deduces Post. 5 thus: 

Suppose that the straight lines making angles with a transversal less than 
two right angles do not meet on the side on which those angles are. 

Then, a fortiori, they will not meet on the other side on which are the 
angles greater than two right angles. 

Hence the straiglit lines will not meet in either direction ; they are there- 
fore parallel. 

But, if so, the angles made by them with the transversal are equal to two 
right angles, by the preceding proposition (= i. 29). 

Thaj^fore the same angles will be both equal to and less than two right 
angles : ▼ 
which is impossible. 

Hence the straight lines will meet 



9o6 



BOOK I 



[l PdST. 5 




IV. Ptolemy lastly enforces his condusion that the strai^t lines will 
meet on the side on which are the emgUs less than two right am^ by recurring 
to the ajbrtiofi step in the forgoing proof. 

Let the angles AFG^ CGJF'm the accompanying figure be together less 
than two right angles. ' 

Therefore the angles BFG^ DGF are greater 
than two right angles. 

We have proved that the straight lines are not 
non-secant 

If they meet, they must meet either towards 
A^ C or towards B^ D. 

(i) Suppose they meet towards B^ A at K. 

Then, smce the angles AFG^ CGFwm less than 
two right angles, and the angles AFG^ GFB are 
equal to two right angles, take away the common angle AFG, and 

the angle CGF is less than the angle BFG; 

that is^ the exterior angle of the triangle JI^FG is less than the interior and 
opposite angle BFG : 
which is impossible. 

Therefore AB, CD do not meet towards B^ D. 

(2) But they do meet, and therefore they must meet in one direction or 
the other : 

therefore they meet towards A^ B^ that is, on the side where are the 
angles less than two right angles. 

The flaw in Rolemy's argument is of course in the part of his proof of 
I. 29 which I have italicised As Produs says, he is not entitled to iissume 
that, if AB^ CD are parallel, whatever is true of the interior angles on one 
side of FG (i.e. that they are together equal to, greater than, or less than, two 
right angles) is necessarily true at the same time of the interior angles on the ' 
other side. Ptolemy justifies this by saying that FA^ GC are no more paralld % 
in one direction tlum FB^ GD are in the other : which is equivalent to the i 
assumption that through any point only one parallel can be drawn to a given , 
straight line. That is, he assumes an equivalent of the very Postulate he b : 
endeavouring to prove. ^ 



Proclus. 



I 



Before passing to his own attempt at a proof, Proclus (p. 368, 26 sqq.) ^ 
examines an ingenious argument (recalling somewhat the famous one about \ 
Achilles and the tortoise) which appeared to show that it was impossible for - 
the lines described in the Postulate to meet j 

Let AB, CD make with ^C the angles BAC, A CD together less than ^ 
two right angles. 

Bisect AC at E and along AB, CD 
respectively measure AF^ CG so that each 
is equal to AE. 

Bisect FG at H and mark off FK, 
GL each equal to FH\ and so on. 

Then AF^ CG will not meet at any 
point on FG ; for, if that were the case, two sides of a triangle would be 
together equal to the third: which is impossible. 



e} \ -h 



fc— 5 



r 



I. Post, s] NOTE OK POSTULATE 5 ao? 

Simflarly, AB^ CD will not meet at any point on KL \ and ''proceeding 
like this indefinitely, joining the non-coincident points, bisecting the lines so 
drawn, and cutting off from the straight lines portions equal to the half of 
these, they say they thereby prove tlmt the straight lines AB^ CD will not 
meet anywhere." 

It is not surprising that Proclus does not succeed in exposing the fallacy 
here (the fact bemg that the process will indeed be endless, and yet the straight 
lines will intersect within a finite distance). But Proclus' criticism contains 
nevertheless something of value. He says that the argument will prove too 
much, since we have only to join AGva order to see that straight lines making 
some angles which are together less than two right angles do in fact meet, 
namely AG^ CO. "Therefore it is not possible to assert, without some definite 
limitation, that the straight lines produced from angles less than two right 
angles do not meet On the contrary, it is manifest that same straight lines, 
when produced from angles less than two right angles, do meet, although the 
argument seems to require it to be proved that this property belongs to aU 
such straight lines. For one might say that, the lessening of the two right 
angles being subject to no limitation, with such and such an amount of 
lessening the straight lines remain non-secant^ but with an amount of lessening 
in excess of this they meet (p. 371, 2—10)." 

[Here then we have the germ of such an idea as that worked out by 
Lobachewsky, namely that the straight lines issuing from a point in a plane 
can be divided with reference to a straight line lying in that plane into two 
classes, ''secant" and "non-secant," and that we may define as parallel the 
two straight lines which divide the secant from the non-secant class.] 

Proclus goes on (p. 371, 10) to base his own argument upon "an axiom 
such as Aristotle too used in arguing that the universe is finite. -For, if from 
one point two straight lines forming an angle Ife produced indefinitely^ the distance 
(&acrTao'c¥, Arist. Scoon/fui) between the said straight lines produ^ indefinitely 
will exceed any finite magnitude. Aristotle at all events showed that, if the 
straight lines drawn from the centre to the circumference are infinite, the 
interval between tliem is infinite. For, if it is finite, it is impossible to 
increase the distance, so that the straight lines (the radii) are not infinite. 
Hence the straight lines, when produced indefinitely, will be at a distance from 
one another greater than any assumed finite magnitude." 

This is a fair representation of Aristotle's argument in De caeh 1. 5, 271 
b 28, although of course it is not a proof of what Proclus assumes as an 
axiom. 

This being premised, Proclus proceeds (p. 371, 24) : 



I. " I say that, if any straight line cuts one of two parallels^ it will cut 
\ the other also, 

"For let AB, CD be parallel, and let EFG cut AB\ I say that it will cut 
k CD also. 
' " For, since BF^ FG are two straight lines from ^ 

one point F^ they have, when produced indefinitely, ^ X^ g 

a distance greater than any magnitude, so that it will ^\ 

J also be greater than the interval between the parallels. Q 

' Whenever therefore they are at a distance from one ^ q 

another greater than the distance between the parallels, 
FG wiU cut CD. 
" Therefore etc" 



ao8 BOOK I 



J 

[l Post. 5 I 




II. *' Having proved this, we shall prove, as a deduction firom it, the 
theorem in question. 

'' For let AB^ CD be two straight lines, and let £^ (idling on them make 
the angles BEF^ DFE less than two right angles. 

''I say that the straight lines will meet on that 
side on which are the angles less than two right 
angles. 

" For, since the angles BEF^ DFE are less 
than two right angles, let the angle HEB be equal 
to the excess of two right angles (over them), and let HE be produced to K. 

''Since then EF falls on KH^ CD and makes the two interior ang^ 
HEF^ DFE equal to two right angles, 

the straight lines HK^ CD are parallel. 

'' And AB cuts KH\ therefore it will also cut CZ>, by what was before 
shown. 

''Therefore AB^ CD will meet on that side on which are the angles less 
than two right angles. 

"Hence the theorem is proved." 

Clavius criticised this proof on the ground that the asdom from which 
it starts, taken from Aristotle, itself requires proofl He points out diat, just 
as you cannot assume that two lines which continually approach one another 
will meet (witness the hyperbola and its asymptote), so you cannot assume 
that two lines which continually diverge will ultimately be so fiLr apart that a 
perpendicular from a point on one let fall on the other will be greater than 
any assigned distance ; and he refers to the conchoid of Nicororiies, whidi 
continually approaches its asymptote, and therefore continually gets fartfier 
away from the tangent at the vertex ; yet the perpendicular from any point on 
the curve to that tangent will always be less than the distance between the 
tangent and the asymptote. Saccheri supports the objection. 

Proclus' first proposition is open to the objection that it assumes that two 
"parallels" (in the Euclidean sense) or, as we may say, two straight lines 
which have a common perpendicular, are (not necessarily equidistant, but) 
so related that, when they are produced indefinitely, the perpendicular from a 
point of one upon the other remains finite. 

This last assumption is incorrect on the hyperbolic hypothesis; the 
"axiom" taken from Aristotle does not hold on the elliptic hypothesis. 

Na^iraddln at-T^si. 

The Persian-bom editor of Euclid, whose date is 1 201— 1274, has three 
lemmas leading up to the final proposition. Their content is substantially as 
follows, the first lemma being apparently assumed as evident 

I. (a) If AB^ CD be two straight lines such that successive perpen- 
diculars, as EF^ GH^ KL^ from points on AB to CD always make with AB 
unequal angles, which are always acute on the side towards B and always 
obtuse on Sie side towards A^ then the lines AB^ 
CD^ so long as they do not cut, approach continually 
nearer in the direction of the acute angles and diverge 
continually in the direction of the obtuse angles, and 
the perpendiculars diminish towards B^ D^ and in- 
crease towards A^ C. 5 — [^ |!| ( r 

ip) Conversely, if the perpendiculars so drawn 
continually become shorter in the direction of B^ Z>, and longer in the 









I. Post. 5] 



NOTE ON POSTULATE 5 



909 




direction of A^ Q the straight lines AB^ CD approach continually nearer in 
the direction of B^ D and diverge continually in the other direction ; also 
each perpendicular will make with AB two angles one of which is acute and 
the other is obtuse, and all the acute angles will lie in the direction towards 
B^ D^ and the obtuse angles in the opposite direction. 

[Saccheri points out that even the first part (a) requires proof. As 
regajrds the converse (p) he asks, why should not the successive acute angles 
made by the perpendiculars with AB^ while remaining acute, become greater 
and greater as the perpendiculars become smaller until we arrive at last at a 
perpendicular which is a common perpendicular to both lines? If that happens, 
all the author's efforts are in vain. And, if you are to assume the truth of the 
statement in the lemma without proof, would it not, as Wallis said, be as 
easy to assume as axiomatic the statement in Post 5 without more ado?] 

II. ^ AC, BD he drawn from the extremities of AB at right angles to it 
and on the same side^ and if AC, BD be made equal to one another and CD be 

joined^ each of the angles ACD, BDC will be rights and 
CD will be equal to AB. 

The first part of this lemma is proved by reductio ad 
absurdum from the preceding lemma. If, e.g., the angle 
ACD is not right, it must eitiier be acute or obtuse. 

Suppose it is acute; then, by lemma i, ^^C is greater 
than BD^ which is contrary to the hypothesis. And so on. 

The angles ACD^ BDC being proved to be right angles, it is easy to 
prove that AB^ CD are equal. 

[It is of course assumed in this "proof" that, if the angle ACD is acute, 
the angle BDC is obtuse, and vice versa.] 

III. /n any triangle the three angles are together equal to two right angles. 
This is proved for a right-angled triangle by means of the forgoing lemma, 

the four angles of the quadrilateral ABCD of that lemma being all right angles. 
The proposition is then true for any triangle, since any triangle can be divided 
into two right-angled triangles. 

IV. Here we have the final "proof" Of Post. 5, Three cases are 
distinguished, but it is enough to show the case where one of the interior 
angles is right and the other acute. 

Suppose AB^ CD to be two straight lines met by FCE making the angle 
ECD a right angle and the angle CEB 
an acute angle. 

Take any point G on £B^ and draw 
GJ£ perpendicular to EC 

Since the angle CEG is acute, the 
perpendicular GH will fall on the side of 
E towards 2?, and will either coincide 
with CD or not coincide with it In the 
former case the proposition is proved. 

If GH does not coincide with CD 
but falls on the side of it towards /^ CZ>, being within the triangle formed by 
the perpendicular and by C£, EG^ must cut EG. [An axiom is here used, 
namely that, if CD be produced far enough, it must pass outside the triangle 
and therefore cut some side, which must be EB^ since it cannot be the 
perpendicular (i. 27), or CE,] 

Lastly, let GHtaM on the side of CD towards E. 
H. B. 14 




9IO BOOK I [l Post. 5 

Along JTCset of[ HK, KL etc, etch equal to EH^ iintU we get the fint 
point of division, as M^ beyond C 

Along GB set off GN^ NO etc, each equal to BG^ until EP ii the mne 
multiple of EG that ^ATis of EH. 

llien we can prove that the perpendiculars from N^ O^ P on EC fiUl on 
the points K^ Z, AT respectively. 

For take the first perpendicular, that from N^ and call it N& 

Draw EQ at right angles to Elf and equal to GB^ and set off 5!i? along 
5A^also equal to GIf. Join Q(r, GE. 

Then (second lemma) the angles EQGf QGIfare right, and QG « Elf. 

Similarly the angles SEG, RGffaxt right, and RG^SH. 

Thus EGQ is one straight line, and the vertically opposite angles NGR^ 
EGQ are equal The angles NRG^ EQG are both right, and NG*^ GE^ by 
construction. 

Therefore (i. 26) EG = GQ; 

whence SIf=^ HE « KH^ and 5 coincides with K. 

We may proceed similarly with tlie other perpendiculars. 

Thus PM is perpendicular to FE. Hence CD^ being parallel to MP and 
within the triangle PME^ must cut EP^ if produced f»x enough. 

John Wallis. 

As is well known, the argument of Wallis (16x6 — 1703) assumed as a 
postulate that, given a figure^ another figure is possible which is similar io the 
gitfen one and of any size whatever. In fiEurt Wdlis assumed this for irbmgles 
only. He first proved (i) that, if a finite straight line is placed on an infinite 
straight line, and is then moved in its own direction as far as we please^ 
it will always lie on the same infinite straight line, (2) that, if an ai^jle be 
moved so that one 1^ always slides along an infinite straight line^ the angle 
will remain the same, or equal, (3) that, if two straight lines, cut by a third, 
make the interior angles on the same side less than two right angles, each 
of the exterior angles is greater than the opposite 
interior angle (proved by means of i. 13). 

(4) If AB, CD make, with A C, the interior 
angles less than two right angles, suppose AC 
(with AB rigidly attached to it) to move along 
AF to the position ay, such that a coincides 
with C If AB then takes the position aj3, a^ lies entirely outside CD (proved 
by means of (3) above). 

(5) With the same hypotheses, the straight line a^, or AB, during its 
motion^ and before a reaches C, must cut the straight line CD. 

!6) Here is enunciated the postulate stated above. 
7) Postulate 5 is now proved thus. 

Let AB^ CD be the straight lines which make, with the infinite straight 
line ACF meeting them, the interior angles 
BA C, DC A together less than two right angles. 

Suppose AC (with AB ri^dly attached to 
it) to move along ACF until AB takes the 
position of o^ cutting CD in ir. 

Then, aCW being a triangle, we can, by 
the above postulate, suppose a triangle drawn 
on the base CA similar to the triangle aCif. 

Let it be ACP. 

[Wallis here interposes a defence of the hypothetical construction.] 





1 



I. Post. 5] NOTE ON POSTULATE s aii 

Thus CP and AP meet at P\ and« as by the definition of similar figures 
the angles of the triangles PCA^ wCa, are respectively equal, the angle PC A 
being equal to the angle irCa and the angle PAC to the angle waC or BAC^ 
it follows that CP^ AP lie on CD^ AB produced respectively. 

Hence AB^ CD meet on the side on which are the angles less than two 
right angles. 

[The whole gist of this proof lies in the assumed postulate as to the 
existence of similar figures ; and« as Saccheri points out, this is equivalent to 
unconditionally assuming the ''hypothesis of the right angle," and consequently 
Euclid's Postulate 5.] 

Gerolamo Saccheri. 

The book EucUdes etb omni ncuvo vindicaius (1733) by Gerolamo Saccheri 
(1667 — 1733), a Jesuit, and professor at the University of Pavia, is now 
accessible (i) edited in German by Engel and Stiickel, DU Theork der 
ParaUtllinien von Euklid bis auf Gauss^ 1895, PP- 4' — ^3^9 ^"^ (?) ^^ ^^ 

(Italian version, abridged but annotated, LEudide em€nd€Uo del P, Gerolamo 
Saccheri^ by G. Boccardini ^Hoepli, Milan, 1904). It is of much greater | 
importance than all the earlier attempts to prove Post 5 because Saccheri 
was the first to contemplate the possibility of hypotheses other than that of 
y Euclid, and to work out a number of consequences of those hypotheses. 
He was therefore a true precursor of Legendre and of Lobachewsky, as 
Beltrami called him (1889), and, it might be added, of Riemann also. For, 
as Veronese observes (Fondamenii di ^ometria^ p. 570), Saccheri obtained 
a glimpse of the theory of parallels m all its generality, while L^endre, 
Lobachewsky and G. Bolyai excluded a /r/V^r/, without knowing it, the ''hypo- 
thesis of the obtuse angle, '^ or the Riemann hypothesis. Saccheri, however, 
was the victim of the preconceived notion of his time that the sole possible 
/ geometry was the Euclidean, and he presents the curious spectacle of a man 
, Uiboriously erecting a structure upon new foundations for the very purpose of 
\ demolishing it afterwards ; he sought for Contradictions in the heart of the 
I systems which he constructed, in order to prove thereby the falsity of his 
hypotheses. 

For the purpose of formulating his hypotheses he takes a plane quadri- 
lateral ABDCy two opposite sides of wluch, AC^ BD^ 
are equal and perpendicular to a third AB. Then the 
angles at C and D are easily proved to be equal On 
the Euclidean hypothesis they are both right angles; 
but apart from this hypothesis they might be both 
obtuse, or both acute. To the three possibilities, which 
I Saccheri distinguishes by the names (i) the hypothesis of 
the right angle^ (2) the hypothesis of the obtuse angle and 
t (3) the hypothesis of the acute angle respectively, there corresponds a certain 
group of theorems ; and Saccheri's point of view is that the Postulate will 
be completely proved if the consequences which follow from the last two 
hypotheses comprise results inconsistent with one another. 

Amons the most important of his propositions are the following : 
(i) Ijthe hypothesis of the right angle^ or of the obtuse an^^ or of the acute 
am^ is proved true in a single case, it is true in ettery other case. (Props, v., 

VI., Vll.) 

(a) According as the hypothesis of the right angle, the obtuse angle, or the 
acute angle is true, the sum of the three .angles of a triangle is equal to, greater 
than, or less than two right angles. (Prop, ix.) 

14—2 



C 


"•-. 

/^<. 




A i 


I 



fx2 BOOK I [lPoct. 5 

(3) From thi exUtence of a sitigk irian^ in tMth ih$ sum of the angUs is 
equal to^ gnaUr ihan^ or less ikon two right augles tki truth of the hypothesis 
of the right angle^ obtuse angle^ or acute em^ respeetivefy fottoms. (Pi^ xv.) 

These propositions involve the foUowing \ If in a single triangle the sum 
of the angles is equal to, greater than^ or less them two right angUs^ then any 
triangle has the sum of its angles equal to^ greater thau^ or less than two right 
angles respectively^ which was (voyed about a century later by Lq;endre for 
the two cases only where the sum is equal to or less than two right angles. 

The proofs are not free from imperfections, as when, in the poob of 
Prop. XII. and the part of Prop. xiii. relating to the hypothesis of the obtuse u 
angle, Saccheri uses Eucl i. t8 depending on l 16, a proposition which is |( 
only valid on the assumption that straight lines are infinite tn length ; for this 1 1 
assumption itself does not bold under the hypothesis of the obtuse angle 
(the Riemann hypothesis). 

The hypothesis of the acute angle takes Saccheri much longer to dispose 
of, and this part of the book is less satisfactory ; but it contains die following 
propositions afterwards established anew by Lobachewsky and Bolyai, viz. : 

(4) Two straight lines in a plane (even on the hypothesis of the acute 
angle) either have a common perpendieular, or must, if produced in one and the 
same direction, either intersect once at a finite distance or at least continueUfy 
approach one another, (Prop, xxiii.) 

(5) In a cluster of rays issuing from a point there exist aluHxys (on the 
hypothesis of the acute angle) two determinate straight lines which sej^araie the 
straight lines which intersect a fixed straight line from those whuh do not . 
intersect it, ending with and including the straight line which has a common 
perpendicular with the fixed straight line. (Props, xxx., xxxi., xxxii.) 

Lambert. 

A dissertation by G.S. Kliigel, Conatuum praecipuorum theoriam parallelarum 
demonstrandi recensio (i 763), contained an examination of some thirty '* demon- 
strations" of Post 5 and is remarkable for its conclusion expressing, apparently 
for the first time, doubt as to its demonstrability and observing that the \ 
certainty which we have in us of the truth of the Euclidean hypothesis is « 
not the result of a series of rigorous deductions but rather of experimental < 
observations. It also had the greater merit that it called the attention of 1 
Johann Heinrich Lambert (1728 — 1777) to the theory of parallels. His 
Theory of Parallels was vrritten in 1766 and published after his death by 
G. Bernoulli and C. F. Hindenburg ; it is reprcxluced by Engel and Stackel ! 
(op. cit. pp. 152 — 208). y 

The third part of Lambert's tract is devoted to the discussion of the same | 
three hypotheses as Saccheri's, the hypothesis of the right an^ being for 
Lambert the first, that of the obtuse angle the second, and that of the acute ] 
angle the thirds hypothesis; and, with reference to a quadrilateral with three ' 
right angles from which Lambert starts (that is, one of the halves into which 
the median divides Saccheri's quadrilateral), Uie three hypotheses are the i 
assumptions that the fourth angle is a right angle, an obtuse angle, or an ' 
acute angle respectively. 

Lambert goes much further than Saccheri in the deduction of new \ 
propositions from the second and third hypotheses. The roost remarkable is { 
the following. , 

The area of a plane triangle^ under the second and third hypotheses, is 
proportional to the eUfference between the sum of the three cmgla and two right 
angles. 






1 

i 









L Post. $] NOTE ON POSTULATE $ ax3 



Thus the numerical expression for the area of a triangle is, under the 
I Mrd hypothesis 

t £L^k\w^A^B^C) (i). 

I and under the sicond hypothesis 

I A = *(^+i?+C-») (a). 

' where ^ is a positive constant 

A remarkable observation is appended (§ 82) : '' In connexion with this it 
seems to be remarkable that the second hypothesis holds if spherical instead of 
plane triangles are taken, because in the former also the sum of the angles is 
gieater than two right angles, and the excess is proportional to the area of the 
triangle. 

''It appears still more remarkable that what I here assert of spherical 
triangles can be proved independently of the difficulty of parallels.' 

This discovery that the second hypothesis is realised on the surface of a 
sphere is important in view of the development, later, of the Riemann 
hypothesis (1854). 

Still more remarkable is the following prophetic sentence: **Iam almost 
inclined to draw the conclusion that the third hypothesis arises with an imaginary 
spherical surfau^^ {ci, Lobachewsky's Gkomitrie ima^naire^ 1^37)* 
^ No doubt Lambert was confirmed in this by the feurt that, in the formula 

(2) above, which, for ^ = f', represents the area of a spherical triangle, if 
r V- I is substituted for r, and r^=^h^ we obtain the formula (i). 

Legendre. 

No account of our present subject would be complete without a full 
reference to what is of permanent value in the investigations of Adrien Marie 
L^endre (i 752-71833) relating to the theory of parallels, which extended over 
, the space of a generation. His different attempts to Drove the Euclidean 
\ hypothesis appeared in the successive editions of his Elkments de Ghmitrie 
I ^m the first (1794) to the twelfth (1823), which last may be said to contain 

I his last word on the subject. Later, in 1833, he published, in the Mhnoires 
de rAcadtmie Royale des Sciences^ xii. p. 367 sqq., a collection of his different 
proofs under the title Rhflexions sur aijjfirentes mani^res de dhnontrer la thhrie 
\ des paraUkks. His exposition brought out clearly, as Saccheri had done, and 
kept steadily in view, the essential connexion between the theory of parallels 
and the sum of the angles of a triangle. In the first edition of the Elkments 
the proposition that the sum of the angles of a triangle is equal to two right 
angUs was proved analytically on the basis of the assumption that the choice 
of a unit of length does not affect the correctness of the proposition to be 
proved, which is of course equivalent to Wallis' assumption of the existence of 
similar figures, A similar analytical proof is given in the notes to the twelfth 
edition. In his second edition L^endre proved Postulate 5 by means of the 
assumption that, ^ven three points not in a straight line^ there exists a arcle 
passing through cUl three. In the third edition (1800) he gave the proposition 
that the sum of the an^s of a triangle is not greater than two ri^ angles] 
this proof, which was geometrical, was replaced later by another, the best 
known, depending on a construction like that of Euclid 1. 16, the continued 
application of which enables any number of successive triangles to be evolved 
in which, while the sum of the angles in each remains always equal to the 
sum of the angles of the original triangle, one of the angles increases and the 
sum of the other two diminishes continually. But Legendre found the proof 
of the equally necessary proposition that the sum of the angles of a triangle is 



JI4 BOOK I [l FdiT. 5 

not less than two right angles to present great diflfculties. He first observed 
that, as in the case of spherical trian^es (in whidi the sum of the angles it 
greater than. two right angles) the excess of the sum of the apgles over two 
right angles is proi)ortional to the area of the triangle, so in the case of 
rectilineal triangles, if the sum of the angles is less than two right angles by « 
certain defidiy th^ deficit will be pioportbnal to the area of the trian^ 
Hence if, starting from a given triangle, we could construct another triaiu^ 
in which the original triangle is contained at least m times, the d^icU of mis 
new triangle will be equal to at least m times that of the original trian^^ so 
that the sum of the angles of the greater triang^ will diminish pro g re s s i vdy 
as m increases, until it becomes lero or negative: which is absurd. The 
whole difficulty was thus reduced to that of the construction of a trianrie 
containing the given triangle at least twice; but the solution of even mis 
simple problem requires it to be assumed (or proved) that tknm^ a ghm 
/0iftt within a given angle less than two4hirds of a ri^ angle we eon eihoays 
draw a straight tine which shall meet both sides of the angle. This is however 
really equivalent to Euclid's Postulate. The proof in the course of whidi the 
necessity for the assumption appeared is as fbUows. 

It is required to prove that the sum of the angles of a triai^ cannot be 
less than two right angles. 

Suppose A is the least of the three angles of a triangle ABC. Apply to 
the oj^posite side BC a triangle DBQ eqiud to 
the triangle ACB, and suoi that the angle 
DBC is equal to the angle ACB^ and the angle 
DCB to the angle ABC ; and draw any straight 
line through D cutting AB^ AC produced in 

If now the sum of the angles of the triangle f 
ABC is less than two right angles, being equal 

to 2^-S say, the sum of the angles of the triangle DBC^ equal to the 
triangle ABC, is also 2^-S. \ 

Smce the sum of the three angles of the remaining triangles DEB, FDC I 
respectively cannot at all events be greater than two right angles [for L^endre's | 
proofs of this see below], the sum of the twelve angles of tibe four triangles in i 
the figure cannot be greater than I 

4^ + (a^««) + (2/?-«X >e. 8^-28. 

Now the sum of the three angles at each of the points B^ C, Dv&iR. i 

Subtracting these nine angles, we have the result that the three angles of ' 
the triangle AEF cannot be greater than 2^-2$. j 

Hence, if the sum of the angles of the triangle ABC is less than two right ^ 
angles by h, the sum of the angles of the laiger triangle AEEis less than two ^ 
right angles by at least 2S. , 

We can continue the construction, making a still larger triangle firom AEE, [ 
and so on. 

But, however small 8 is, we can arrive at a multiple 2*8 which shall exceed : 
any given angle and therefore 2B itself; so that the sum of the three angles I 
of a triangle sufficiently laige would be zero or even less than zero : which is 
absurd. 

Therefore etc. 

The difficulty caused by the necessity of making the above-mentioned 
assumption made Legendre abandon, in his ninth edition, the inethod of the 




1 



t 



I. Post, s] NOTE ON POSTULATE 5 215 



editions from the third to the eighth and return to Euclid's method pure and 
simple. 
) But again, in the twelfth, he returned to the plan of constructing any 

I number of successive triangles such that the sum of the three angles in all of 
I them remains equal to the sum of the three angles of the original triangle, 
but two of the angles of the new triangles become smaller and smaller, while 
the third becomes larger and laiger ; and this time he claims to prove in one 
proposition that the sum of the three angles of the original triangle is r^a/ to 
two right angles by continuing the construction of new triangles indefinitely 
and compressing the two smaller angles of the ultimate triangle into nothing, 
while the third angle is made to become a flat angle at the same time. The 
construction and attempted proof are as follows. 

Let ABC be the given triangle ; let AB be the greatest side and BC the 
least ; therefore C is the greatest angle and A the least. 

From A draw AD to the middle point of BC^ and produce AD to C", 
making AC equal to AB. 

Produce AB to B^ making AB equal to twice AD. 

The triangle ABC is then such that the sum of its three angles is equal 
to the sum of the three angles of the triangle ABC. 




For take ^ A' along AB equal to AD^ and join CK. 

Then the triangles ABD^ ACK have two sides and the included angles 
respectively equal, and are therefore equal in all respects ; and CK is equal to 
BD or DC 

Next, in the triangles BCK^ ACD, the angles BKC\ ADC are equal, 
being respectively supplementary to the equal angles AKC\ ADB\ and the 
two sides about the equal angles are respectively equal; 

therefore the triangles BCK^ A CD are equal in all respects. 

Thus the angle ACB is the sum of two angles respectively equal to the 
angles B^ C of &e original triangle ; and the angle A in the original triangle 
is the sum of two angles respectively equal to the angles at A and B in the 
triangle ABC. 

It follows that the sum of the three angles of the new triangle ABC is 
equal to the sum of the angles of the triangle ABC. 

Moreover, the side AC\ bein^ equal to AB^ and therefore greater than 
AC^ is greater than BC which is equal to AC. 

Hence the angle CAB is less than the angle ABC ; so that the angle 
CAB is less than ^A^ where A denotes the angle CAB of the original 
triangle. 

[It will be observed that the triangle ABC is really the same triangle as 
the triangle AEB obtained by the construction of Eucl. i. 16, but differendy 
placed so that the longest side k'es along AB.] 

By taking the midcUe point D of the side B'C and repeating the same 
construction, we obtain a triangle A B'C' such that (i) the sum of its three 
angles is equal to the sum of the three angles of ABCf (2) the sum of the 




I 



9i6 BOOK I [lFobt. 5 

two angles C'AB\ AB'C* is equal to the aii|^ CAB in tbe preceding 
triangle, and is therefore less than \A^ and (3) the an|^ CAB* is kas than 
half &e angle CAB>^ and therefore ie» than ^A. 

Continuing in this way, we shall obtain a triangle Ahc such that the sum of 

two angles, those at A and b^ is less than -^A^ and the an|^ at r is greater 

than the corresponding angle in the preceding triangle. 

If, L^endre argues, the construction be continued indefinitely so that 

-^A becomes smaller than any assigned angles the point c ultimately lies on 

Aby and the sum of the three ai^es of the triangle (which is equal to the sum j 
of the three angles of the original triangle) becomes identical with the angle | 
at r, which is then 2^ flat angle, and therdrore equal to two right angles. 

This proof was however shown to be unsound (in remct Si the final K 
inference) by J. P. W. Stein in Gergonne's Afmales de MoiUmaHfuis xv., 
1824, pp. 77—79- 

We will now reproduce shortly the substance of the theorems of Legendre 
which are of the most permanent value as not depending on a particular 
hypothesis as regards parallels. 

I. 7%e sum of the thre^ anf^ cf a irian^ cannci be greater than iW0 
right atigks. 

This Legendre proved in two ways. 

(i) First proof (m the third edition of the iUments). 

Let ABC be the given triangle, and ACJ9l straight line. 

Make CE equal to AC, the angle DCE equal to the angle BAC^ and DC 
equal to AB. Join DE. 

Then the triangle DCE is equal to the triangle BAC in all respects. 

If then the sum of the three angles of the triangle ABC is greater than 

{ 

{ 



2Ey the said sum must be greater than the sum of the angles BCA^ BCD, 
DCE, which sum is e^al to 2^. 

Subtracting the equal angles on both sides, we have the result that i 

the angle ABC is greater than the angle BCD. ( 

But the two sides AB, BC of the triangle ABC are respectively equal to i 
the two sides DC, CB of the triangle BCD. 

Therefore the base AC is greater than the base BD (EucL i. 24). 

Next, make the triangle FEG (by the same construction) equal in all ^ 
respects to the triangle BAC or DCE; and we prove in the same way that * 
CE (or AC) is greater than DF. 

And, at the same time, BD is equal to DF^ because the angles BCD, 
DEFMxe equal. 

Continuing the construction of fiirther triangles, however small the 
difference between AC and BD is, we shall ultimately reach some multiple j 







' 1.P0ST. s] NOTE ON POSTULATE $ »i7 

I of this difference, represented in the figure by (say) the difference between 
^. the straight line A/ and the composite line BDFHK^ which will be greater 
than any assigned length, and greater therefore than the sum of AB vcAJK, 

Hence, on the assumption that the sum of the angles of the triangle ABC 
is greater than 2^, the broken line ABDFHKJ may be less than the straight 
line AJ\ which is impossible. 

Therefore etc 

(2) Proof substituted later. 

If possible^ let 2^ -f a be the sum of the three angles of the triangle ABC^ 
of which A is not greater than either of the 
others. 

Bisect BC at H^ and produce AH to Z>, 
making HD equal to AH\ join BD. 

Then the triangles AHC^ DHB are equal in 
all respects (i. 4) ; and the angles CAH^ ACHzx^ 
respectively equal to the angles BDH^ DBH. 

It follows that the sum of the angles of the 
triangle ABD is equal to the sum of the angles of the original triangle, i.e. 
to 2^ + a. 

And one of the angles DAB^ ADB is either equal to or less than half the 
angle CAB. 

Continuing the same construction with the triangle ADB^ we find a third 
triangle in which the sum of the angles is still 2^ + 0, while one of them is 
equal to or less than ( l CAB)I^. 

Proceeding in this way, we arrive at a triangle in which the sum of the 
angles is 2^ + a, and one of them is not greater than ( l CAB)l2\ 

And, if n is sufficiently large, this will be less than a ; in which case we 
should have a triangle in which two angles are together greater than two right 
angles : which is absurd. 

Therefore a is equal to or less than zero. 

(It will be noted that in both these proofs, as in Eucl. i. 16, it is taken for 
granted that a straight One is infinite in length and does not return into itself, 
which is not true under the Riemann hypothesis.) 

II. On the assumption that the sum of the angles of a triangle is less 
than two right angles, if a triangle is made up of two others^ the ^^ deficit** of the 
former is equal to the sum of the ^^ deficits** of the others. 

In fact, if the sums of the angles of the component triangles are 2^ - a, 
i/i-p respectively, the sum of the angles of the whole triangle is 

(2^-a) + (2-^-/9)-2^ = 2-^-(a + /9). 

III. If the sum of the three angles of a triangle is equal to two ri^t 
angles^ the same is true of all trian^es obtained by subdividing it by straight 
lines drawn from a vertex to meet the opposite side. 

Since the sum of the angles of the triangle ABC is equal to 2^, if the 
sum of the andes of the triangle ABD were 2^- a, it 
would follow that the sum of the angles of the triangle 
ADC must be 2^ + a, which is absurd (by I. above). 

IV. If in a triangle the sum of the three angles is 
equal to two right angles ^ a quadrilcUeral can cUways be 
constructed with four right angles and four equal sides 
exceeding in length any assigned rectilineal segment. 

Let ABC be a triangle in which the sum of the angles is equal to two 




si8 



BOOK I 



[lFobt. 3 



Tight angles. We can assume ABC to be an isaseeies rigfit-am^ki triaiude 
because we can reduce the case to this by makiiig subdivisions of ABCvr/ 
straight lines through vertices (as in Prop, iil above). 

Taking two equal triangles of this kind and placing their hypotennies 
together, we obtain a quadrilateral with four right angles and four equal 
sides. 

Putting four of these quadrilaterals together, we obtain a new quadrilateral 
of the same kind but with its sides double of those of the first quadrilateral 

After n such operations we have a quadrilateral with four rigt^t angles and 
four equal sides, each being equal to s* times the side AB. 

The diagonal of this qiuidrilateral divides it into two equal isosceles ri^t- 
angled triangles in each of whicb tibe sum of the angles is equal to two ng^t 
angles. 

Consequently, from the existence of one triansle in which the sum of the 
three angles is equal to two right angles it follows ttiat there exists an isoscetes 
right-angled triangle widi sides greater than any assigned redDineal segment 
and such that the sum of its three angles is also equal to two right ang^ 

V. If the sum of the three angies of one triangle is equal to twoo rigkt 
angles^ the sum of the three angks of any other triauffe is also ofual to twoo 
right angies. 

It is enough to prove this for a rigki-anf/ed triangle, since any triangle can 
be divided into two right-angled triangles. 

Let ABC be any right-angled triangle. 

If then the sum of the angles of any one 
triangle is equal to two right angles, we can 
construct (by the preceding Prop.) an isosceles 
right-angled triangle with the same property and 
with its perpendicular sides greater than those of 
ABC 

Let ABC be such a triangle, and let it be 
applied to ABC^ as in the figure. 

Applying then Prop. in. above, we deduce 
first that the sum of the three angles of the 
triangle ABC is equal to two right angles, and 

next, for the same reason, that the sum of the three angles of the original 
triangle ABC is equal to two right angles. 

VI. If in any one triangle the sum of the three an^s is less than two 
right angles, the sum of the three angjUs of any other triangle is also less than 
two right angles. 

This follows from the preceding theorem. 

(It will be observed that the last two theorems are included among those 
of Saccheri, which contain however in addition the corresponding theorem 
touching the case where the sum of the angles is greater than two right 
angles.) 

We come now to the bearing of these propositions upon Euclid's Postulate 
5 ; and the next theorem is 

VII. If the sum of the three angles of a trian^e is equal to two ri^ 
angles^ through any point in a plane there can only be drawn one parallel to a 
givem straight line. 




I 




I. Post. 5] NOTE ON POSTULATE 5 319 

For the proof of this we require the following 

Lemma. // is a/ways possible^ through a point P, to draw a straight line 
which shall mahe^ with a gh^en straight line (r), an angle less than any assigned 
angle. 

Let Q be the foot of the perpendicular from i'upon r. 

Let a segment QR be taken on r, 



on either side of Q, such that QR is 
equal to PQ, 

Join PR^ and mark off the segment 
RR' equal to PR ; join PR\ 

If m represents the angle QPR or 
the angle QRP^ each of the equal 
angles RPR\ RR'P is not greater 
than a>/2. 

Continuing the construction, we obtain, after the requisite number of 
operations, a triangle /^Rn-i Rn in which each of the equal angles is equal to 
or less than m/a*. 

Hence we shall arrive at a straight line PR,^ which, starting from /'and 
meeting r, makes with r an angle as small as we please. 

To return now to the Proposition. Draw from P the straight line s 
perpendicular to PQ. 

Then any straight line drawn from P which meets r in ^ will form equal 
angles with r and x, since, by hypothesis, the sum of the angles of the triangle 
PqR is equal to two right angles. 

And since, by the I^mma, it is always possible to draw through i'strai^t 
lines which form with r angles as small as we please, it follows that all the 
straight lines through P^ except x, will meet r. Hence s is the only parallel 
to r that can be drawn through P. 

The history of the attempts to prove Postulate 5 or something equivalent 
has now been brought down to the parting of the ways. The further 
developments on lines independent of the Postulate, bc^ning with 
Schweikart (1780 — 1857), Taurinus (1794 — 1874)^ Gauss (1777 — 1855), 
Lobachewsky (1793 — 1856), J. Bolyai (1802— 1860) , Riemann (1826— 1866), 
belong to the history of non-Euclidean geomefry, which is outside the scope 
of this work. I may refer the reader to the full article Snlla teoria delie 
parallele e suite geometrie non-euclidee by R. Bonola in Questioni riguardanti 
I la geometria elementare^ 1900, of which I have made considerable use in the 

[' above, to the same author's La geometria non-eudidea^ Bologna, 1906, to the 
first volume of Killing's Einfiihrung in die Grundlagen der Geometrie^ 
Paderbom, 1893, to P. Mansion's Premiers principes de nUtaghmktrie^ and 
) P. Barbann's La gtomttrie non-Euclidienne^ Paris, 1902, to the historical 
' summary in Veronese's Fondamenti di geometria^ 1891, p. 565 sqq., and (for 
original sourc^ to Engel and Stackel, Die Theorie der Parallellinien von 
Euhlid bis auf Gauss^ 1895, ^^^ Urkunden %ur Geschichie der nicht-Euhlidischen 
Geometriey i. (Lobachewsky), 1899, and 11. (Wolfgang und Johann Bolyai). 
I will only add that it was Gauss who first expressed a conviction that the 
Postulate could never be proved ; this he stated distinctly, first in a review in 
the Gottingische gdehrte Anseigen, 20 A pril 1816, and secondly in a letter to 
Bessel of 27 Januaiy, 1829. The actual inaemonstrability of the Postulate was 
proved by Beltrami (1868) and by Hoiiel {Note sur Timpossibiliti de dSmontrtr 
par une construetion plane le prindpe de la thkorie des paralUles dit Postulatum 
d*Euclide in Battaglini's Giomale di matematiche^ viii., 1870, pp. 84 — 89). 



»o BOOK I [vTon.s 

Alternatives for Postulate 5. 

It may be convenient to odiect here a few of the more ndewortby 
substitutes which have from time to time been fbnnally suggested or tac^ 
assumed. 

(i) Through a ghen paini mfy 9m parallel €an he drawn io a given 
straight line or, Two straight Knes which interseet one another eamnat hath he 
parallel to one and the same straight Hne. 

This is commonly known as '' Playfair's Axiom," but it was of coorie not 
a new discovery. It is distinctly stateo in Produs* note to EucL i. 31. 

(i a) If a straight line interseei one rf twoo parallels^ it wHl inieruet the 
other also (Proclus). 

(i b) Straight lines parallel to the same straight line are paralM to one 
another. 

The forms (i a) and (i b) are exactly equivalent to (i). 

(s) There exist straight Unes everywhirt oguidistemt firmn on€ emctker 
(Posidonius and Geminus); with which may be compared Produs* ticit 
assumption that Parallels remain^ throngkout their lenf^n^ at a finite distama 
from one another. 

(3) There exists a trian^e in whieh the sum of the three an^ iseqmUto 
two right angles (L^endre). 

(4^ Given any figure^ there exists a figure similar to it of any siu wepleau 
(Walhs, Cftmot, Laplace). 

Saccheri points out that it is not necessary to assume so much, and that it 
is enough to postulate that there exist two unequal triemj^ with eqmd angles. 

(5) Through any point within an angle less than two4hirds of a right at^ 
a straight line can always be drawn which meets both sides of the anj^e 
(Legendre). 

With this may be compared the similar axiom of Lorenz {Grundriss der 
reinen und angewandten Mathematih^ 1791): Every straight line through a 
point within an angle must meet one of the sides of the angle, 

(6) Given any three points not in a straight line^ there exists a circle pcusing 
through them (Le^ndre, W. Bolyai). 

(7) . ^* If I could prove that a rectilineal triangle is possible the content of 
which is greater than any given area^ I cm in a position to prove perfectly 
rigorousfy the whole of geometry^ (Gauss, in a letter to W. Bolyai, 1799). 

Cf. the proposition of Legendre numbered iv. above, and the axiom of 
Worpitzky: TTiere exists no tricmgle in which every angU is cu small as we 
please. 

(8) If in a quadrilateral three angles are right angles^ the fourth angle is 
a right angle also (Clairaut, 1741). 

(9) If two straight lines are parcUld^ they are figures opposite to (or the 
reflex of) one another with respect to the middle points of all their transversal 
segments (Veronese, Elementi^ 1904). 

Or, Two parallel straight lines intercept, on every transversal which passes 
through the middle point of a segment included betwun them^ another segment 
the middle point of which is the middle point of the first (Ingrami, ElemenH, 
>904). 

Veronese and Ingrami deduce immediately Playfair's Axiom, 






;i 



NOTES ON THE COMMON NOTIONS »ii 



AXIOMS OR COMMON NOTIONS. 



\ 

' In a paper Sur Pauthentkiti des axiomes ttEuclide in the Bulletin des 

sciences nuUMmatiques et ustronamiques^ a* s^r. viii., 1884, p. 162 sqq., Paul 

' Tannery maintained that the Common Notions (including the first three) were 
not in Euclid's work but were interpolated later. The following are his main 
aiguments. (i) If Euclid had set about distin^ishing between indemon- 
stmble principles (a) common to all demonstrative sciences and (b) peculiar 
to geometry, he would, says Tannery, certainly not have placed the common 
principles second and the special principles (the Postulates) first. (2) If the 
Common Notions are Euclid's, thb designation of them must be his too ; for he 
must have used some name to distinguish them from the Postulates and, if he 
had used another name, such as Axioms^ it is impossible to imagine why that 
name was changed afterwards for a less suitable one. The word hn^wa 
{notion\ ^ys Tannery, never signified a notion in the sense of a proposition^ 
but a notion of some odjat ; nor is it found in any technical sense in Plato 
and Aristotle. (3) Tannery's own view was that the formulation of the 
Common Notions dates from the time of Apollonius, and that it was inspired 
by his work relating to the Elements (we know from Proclus that Apollonius 
tried to prove the Common Notions), This idea, Tannery thought, was 

J confirmed by a " fortunate coincidence " furnished by the occurrence of the 
word Hyyota (notion) in a quotation by Proclus (p. 100, 6): ''we shall agree 
with Apollonius when he says that we have a nolion (lirocay) of a line when 
we order the lengths, only, of roads or walls to be measured." 

In reply to argument (i) that it is an unnatural order to place the purely 
geometrical Postulates first, and the Common Notions^ which are not peculiar 
to geometry, last, it may be pointed out that it would surely have been a still 
more awkward arrangement to give the Definitions first and then to separate 
from them, by the interposition of the Common Notions^ the Postulates, which 
are so dosely connected with the Definitions in that they proceed to postulate 
the existence of certain of the things defined, namely straight lines and circles. 

(2) Though it is true that wvow, in Plato and Aristotle is generally a 
notion of an object^ not of difact or proposition, there are instances m Aristotle 
where it does mean a notion of a fiact : thus in the Eth, Nic, ix. 11, ii7i a32 
he speaks of ''the notion (or consciousness) that friends sympathise^ (17 &voia 
rw (TvraXyciy rovs ^cXovs) and again, b 14, of "the notion (or consciousness) 
that they are pleased at his good fortune." It is true that Plato and Aristotle 
do not use the word in a technical sense ; but neither was there apparently in 
Aristotle's time any fixed technical term for what we call "axioms," since he 
speaks of them variously as " the so-called axioms in mathematics," " the so- 
odled common axioms," "the common (things)" (ra xocm), and even "the 
common opinions " (jcoiml 80^1). I see therefore no reason why Euclid should 
not himself have given a technical sense to " Common Notions," which is at 

I ' least a distinct improvement upon "common opinions." 

(3) The use of hn^wa in Proclus' quotation from Apollonius seems to me 
.. to be an unfortunate, rather than a fortunate, coincidence from Tannery's point 

of view, for it is there used precisely in the old sense of the notion of an 
object (in that case a line). 

No doubt it is difficult to feel certain that Euclid did himself use the term 
Common Notions^ seeing that Proclus' commentary generally speaks of Axioms. 
But even Proclus (p. 194, 8), after explaining the meanmg of the word 
"axiom," first as used by the Stoics, and secon<fiy as used by "Aristotle and 



t32 • BOOK I [lCMi 

the geometers," goes on to say : ''For in their view (that of Aristotk and the 
geometers) axiom and common naium are the same thing." This, as it seems 
to me, may be a sort of apology for using the word ''axiom " exdusivdy in 
what has ^one before, as if Prochis had suddenly bethou^t himself that he 
had descnbed both Aristotle and the geometers as using the one tenn 
"axiom," whereas he should have said that Aristotle spoke of "axiomsi* while 
"the geometers" (in fact EuclidX though meaning the same thing, odled them 
Common Notions. It may be for a like reason that in another passage (p. 76, 
16X after quoting Aristotle's view of an "axiom," as distinct from a portukfr 
and a hypothesis, he proceeds : " For it is not by virtue of a iommom noHon 
that, without being taught, we preconceive the circle to be such and such a 
figure." If this view of the two passages just (quoted is correct, it would 
strengthen rather than weaken the case for the genwneness of Common NoHoms 
as the Euclidean term. 

Again, it is clear from Aristotle's allusions to the "common axioms in 
mathematics " that more than one axiom of this kind had a place in Uie text- 
books of his day ; and as he constantly quotes the particular axiom that, y 
eqnais be taken from equals^ the remainders are equals which is Eudid*^ Common 
Notion ^ it would seem that at least die first three Common Notions were 
adopted by Euclid from earlier text-books. It is, besides, scarcely credifa3e 
that, if the Common Notions which ApoUonius tried to Drove had not been 
introduced earlier (e.g. by Euclid), they would then have been interpolatad as 
axioms and not as propositions to be proved. The line taken by Apollonius 
is much better explained on the assumption that he was directly attacking 
axioms which he found already admitted mto the Elements. 

Proclus, who recognised die five Common Notions given in the text, warns 
us, not only against the error of unnecessarily multiplying the axioms, but 
against the contrary error of reducing their number unduly (p. 196, 15), "as 
Heron does in enunciating three only; for it is also an axiom that the whole is 
greater than the party and mdeed the geometer employs this in many places for 
his demonstrations, and again that things which coinade are egucU** 

Thus Heron recognised the first three of the Common Notions ; and this 
foct, together with Aristotle's allusions to "common axioms*' (in the plural), 
and in particular to our Common Notion 3, may satisfy us that at least the first 
three Common Notions were contained in the Elements as they left Euclid's 
hands. 

Common Notion i. 

Ta vf a^rf ura jcal o^Xi/Xotf joriy Zero. 

Things which are equal to the same thing are also equal to one another, 

Aristotle throughout emphasises the fact that axioms are self-evident truths, 
which it is impossible to demonstrate. If, he says, any one should attempt to 
prove them, it could only be through ignorance. Aristotle therefore would 
undoubtedly have agreed in Proclus' strictures on Apollonius for attempting 
to prove the axioms. Proclus gives (p. 194, 25), as a specimen 
of these attempted proofs by Apollonius, that of the first of the 
Common Notions. " Let A be equal to Bj and the latter to C; 
I say that A is also equal to C. For, since A is equal to £/\t A B 
occupies the same space with it ; and since B is equal to C, it 
occupies the same space with it. 

Therefore A also occupies the same space with C" 

Proclus rightly remarkis (p. 194, 22) that "the middle term is no more 



. 
► 



I. CM 1—3] NOTES ON COMMON NOTIONS 1—3 iti 

intelligible (better known, yvwptfuirtfiov) than the conclusion, if it is not 
actually more disputable." Again (p. 195, 6), the proof assumes two things, 
(i) that things which "occupy the same space" (roiros) are equal to one 
another, and (2) that things which occupy the same space with one and the 
same thing occupy the same space with one another ; which is to explain the 
obvious by something much more obscure, for space is an entity more 
unknown to us than the things which exist in space. 

Aristotle would also have objected to the proof that it is partial and not 
general (xotfoXov), since it refers only to things which can be supposed to 
occupy a space (or take up room), whereas the axiom is, as Proclus says 
(p. 196, i), true of numbers, speeds, and periods of time as well, though of 
course each science uses axioms in relation to its own subject-matter only. 

Common Notions 2, 3. 

2. Kai iav urocc icra irpo(rr€0j, ra oXa iartv urcu 

3. Koi iav airo icrwy ccra dflHup€6i, ^^ KaroXciird/Acva ifrriv icro. 

2. If equals be added to equals^ the wholes are equal, 

3. If equals be subtracted from equals^ the remainders are equal. 

These two Common Notions are recognised by Heron and Proclus as 
genuine. The latter is the axiom which is so favourite an illustration with 
Aristotle. 

Following them in the mss. and editions there came four others of the 
same type as i — ^^3. Three of these are given by Heiberg in brackets ; the 
fourth he omits altogether. 

The three are : 

(a) If equals be adtled to unequals^ the wholes are unequal, 

iff) Things which are double of the same thing are equal to one another, 

(c) Things which are halves of the same thing are equal to one anotlur. 
The fourth, which was placed between {a) and (^), was : 

(d) If equals be subtracted from unequals^ the remainders are unequal, 

Proclus, in observing that axioms ought not to be multiplied, indicates 
that all should be rejected which follow ^m the five admitted by him and 
appearing in the text above (p. 155). He mentions the second of those just 
quoted {p) as one of those to be excluded, since it follows from Common 
Notion I. Proclus does not mention (a), (r) or (d)\ an-NairizI gives (a), {fl\ (b) 
and (^), in that order, as Euclid's, adding a note of Simplicius that '' three 
axioms (sententiae acceptae) only are extant in the ancient manuscripts, but 
the number was increased in the more recent" 

(a) stands self-condemned because " unequal " tells us nothing. It is easy 
to see what is wanted if we refer to i. 17, where the same angle is added to a 
greater and a Uss^ and it is inferred that the first sum is greater than the second. 
So far however as the wording of (a) is concerned, the addition of equal to 
greater and less might be supposed to produce less and greater respectively. If 
therefore such an axiom were given at all, it should be divided into two. 
Heiberg conjectures that this axiom may have been taken from the commentary 
of Pappus, who had the axiom about equals added to unequals quoted below 
{ey^ if so, it can only be an unskilful adaptation of some remark of Pappus, for 
his axiom {e) has some point, whereas (a) is useless. 

As regards (b\ I agree with Tannery in seeing no sufficient reason why, if 



t 

J 

i 



224 BOOK I [l C.JK 

we reject it (as we certainly must), the words in i. 47 '^But things irtiich aie 
double of equals are equal to one another" should be condemned as an 
interpolation. If they were interpoUted, we should have eipected to find the 
same interpolation in i. 42, where the axiom is ioMy assumed. I diink 
it quite possible that Euclid may have inserted such words in one case and 
left them out in another, without necessarily implying either that he was 
quoting a formal Common Naium of his own or that he had mi inchided 
among his Common Notions the particular fiu:t stated as obvious. 

The corresponding axiom (r) about the kahes of a^uals can hardly be 
genuine if {b) is not, and Produs does not mention it Tannery acutdy 
observes however that, when Heibeig, in i. 37, 38, brsckets words statiiw that 
^'the halves of equal things are equal to one another** on the ground that 
axiom {f) was interpolated (althou^ before Theon's time), and explains that 
Euclid used Common Notion ^ in making his inference, he is clearly mistaken. 
For, while axiom (b) is an obvious inference from Common Notion s, axiom lA 
is not an inference from Common Notion 3. Tannery says, in a note, that (q 
would have to be established by redtutio ad atsunhtm with the help of axiom 
{b\ that is to say, of Common Notion s. But, as the hypothesis in ttie roAuHo 
adabsurdum would be that one of the halves ispraier than the other, and it 
would therefore be necessary to prove that the one whole is gnoUr than the 
other, while axiom {b) or Common Notion % only refers to oqfuds^ a little 
argument would be necessary in addition to the refimnce to Common Notion s. 
I diink Euclid would not have gone through this process in order to prove (^), 
but would have assumed it as equally obvious with (by 

Proclus (pp. 197, 6— 19S, 5) definitely rejects two other axioms of the 
above kind given by Pappus, observing that, as they follow fimn the {pmine 
axioms, they are rightly omitted in most copies, although Pappus said that 
they were '' on record " with the others (w¥aj¥aypau^wBai) : 

(e) If unequais be added to equals^ the difference between the wholes is equal 
to the difference between the added parts \ and 

(/) If equals be added to unequais^ the difference between the wholes is equal 
to the difference between the original unequcUs. 

Proclus and Simplidus (in an-Nairi2l) give proofs of both. The proof of 
the former, as given by Simplicius, is as follows : 

Let A£, CD be equal magnitudes ; and let E£, FD be £ 
added to them respectively, EB being greater than FD. q 

I say that AE exceeds CF by the same difference as that by 
which BE exceeds DF B 

Cut ofif from BE the magnitude BG equal to DF. 

Then, since AE exceeds AG by GE^ and AG is equal to CF 
and BG to DF, 

AE exceeds CF by the same difference as that by which BE 
exceeds DF. I 

Common Notion 4. ^ 

Kal ra i^apfi6(,oyra cir* oXXi^Aa ura aXXi/Xoif coTiK. * 

7%ings whkh coincide with one another are equal to one another. 
The word c^kpfuSfctv, as a geometrical term, has a different meaning | 
according as it is used in the active or in the passive. In the passive, | 
i^apfiHw^ai, it means "to be applied to" without any implication that the 
app^ed figure will exactly fit, or coincide with, the figure to which it is applied; 
on the other hand the active i^apfioC€tv is used intransitively and means ''to 



T 



r 



I. C N. 4] NOTES ON COMMON NOTIONS 2—4 ^^5 

fit exactly," "to coincide with." In Euclid and Archimedes i^apyuUgmif is 
constructed with ciri and the accusative, in Pappus with the dative. 

On Common Notion 4 Tannery observes that it is incontestably geometrical 
in character, and should therefore have been excluded from the Common 
Notions; again, it is difficult to see why it is not accompanied by its converse, 
at all events for straight lines (and, it might be added, angles also), wluch 
Euclid makes use of in i. 4. As it is, says Tannery, we have here a definition 
of geometrical equality more or less sufficient, but not a real axiom. 

It is true that Proclus seems to recognise this Common Notion and the next 
as proper axioms in the passage (p. 196, 15 — 21) where he says that we should 
not cut down the axioms to the minimum, as Heron does in giving only three 
axioms; but the statement seems to rest, not upon authority, but upoo an 
assumption that Euclid would state explicitly at the b^inning all axioms 
subsequently used and not reducible to others unquestionably included. Now 
in I. 4 this Common Notion is not quoted ; it is simply inferred that " the base 
BC will coincide with EF^ and will be equal to it." The position is therefore 
the same as it is in regard to the statement in the same propositipn that, *Hf... 
the base BC does not coincide with EF^ two straight lines will enclose a sface : 
which is impossible " ; and, if we do not admit that Euclid had the axiom tiiat 
" two straight lines cannot enclose a space," neither need we infer that he had 
Common Notion 4. I am therefore inclined to think that the latter is mcn-e 
likely than not to be an interpolation. 

It seems clear that the Common Notion, as here formulated, is intended 
to assert that superposition is a legitimate way of proving the equality of two 
figures which have the necessary parts respectively equal, or, in other wqrIs, 
to serve as an axiom of congruence. 

The phraseology of the propositions, e.g. i. 4 and i. 8, in which Eodid 
employs the method indicated, leaves no room for doubt that he regarded one 
figure as actually moved and placed upon the other. Thus in i. 4 he ays, 
"The triangle ABC being applied (c^op/tofo/Acvov) to the triangle DEF^wcA 
the point A being plaad {jtM^ivwi) upon the point />, and the straight line 
AB on DE^ the point B will also coincide with E because AB is equd to 
DE''\ and in i. 8, " If the sides BA^ AC do not coincide with ED, DF, but 
fall beside them (take a different position, iropaXXa^ouo-iF), then " etc At the 
same time, it is clear that Euclid disliked the method and avoided it whereirer 
he could, e.g. in i. 26, where he proves the equality of two triangles which have 
two angles respectively equal to two angles and one side of the one equd to 
the corresponding side of the other. It looks as though he found the method 
handed down by tradition (we can hardly suppose that, if Thales proved tiiat 

) the diameter of a circle divides it into two equal parts, he would do so by any 
other method than that of superposition), and followed it, in the few cues 
where he does so, only because he had not been able to see his way to a 

i satisfactory substitute. But seeing how much of the Elements depends on l 4, 
directly or indirectly, the method can hardly be regarded as being, in Eodid, 
of only subordinate importance ; on the contrary, it is fundamental Nor, as 

< a matter of fact, do we find in the ancient geometers any expression of doubt 
as to the legitimacy of the method. Archimedes uses it to prove that any 

. spheroidal figure cut by a plane through the centre is divided into two equal 

I parts in respect of both its surface and its volume; he also postulates in 
Equilibrium of Planes i. that " when equal and similar plane figures coincide 
if applied to one another, their centres of gravity coincide also." 

j Killing (Einfuhrung in die Grundlagen der Geometrie^ 11. pp. 4, 5) 

H. E. 15 



236 BOOK I [lCMa 

contrasts the attitude of the Greek geometers with that of the philosc^riieni 
who, he says, appear to have agreed in banishing motion from geometry 
altogether. In support of this he refers to the view frequently expressed l^ 
Aristotle that mathematics has to do with immovable objects (aiciri|raX and that 
only where astronomy is admitted as part of mathematical science is motion 
mentioned as a subject for mathematics. Cf. MiU^h. 989 b 32 ''For madie- 
matical objects are among things which exist i^Mut from motion, except sudi 
as relate to astronomy"; Metapk. 1064 a 30 ''Physics deals with things 
which have in themselves the principle of motion; mathematics is a theoretical 
science and one concerned with things which are siatumary (/imirra) but not 
separable" (sc. from matter^; in Physics 11. 2, 193 b 34 he speaks of the 
subjects of mathematics as "m thought separable from motion." 

But I doubt whether in Aristotle's use of the words "immovaUe," "widi- 
out motion" etc as applied to the subjects of mathematics diere is any 
implication such as Killing supposes. We arrive at mathematical concepts 
by abstraction from material objects ; and just as we, in thought, eliminate 
the matter, so according to Aristotle we diminate the attributes of matter as 
such, e.g. qualitative clmnge and motion. It does not appear jto me that die 
use of " immovable " in the passages referred to means more than this. I do 
not think that Aristotle would have r^purded it as illegitimate to mom z. 
geometrical figure from one position to another; and I infer this from a 
passage in De caelo iii. i where he is criticising "those who make up every 
iKxly that has an origin by putting together pianes^ and resolve it again into 
planesJ' The reference must be to the TimaeHS (54 B sqq.) where Plato 
evolves the four elements in this way. He begins with a right-angled triangle 
in which the hypotenuse is double of the smaller side; six of these put together 
in the proper way produce one equilateral triangle: Making solid uigles with 
(a) three, (b) four, and (c) five of these equilat^al triangles respectively, and 
taking the requisite number of these solid angles, namely four of (tf), six of (b) 
and twelve of (c) respectively, and putting them together so as to form regular 
solids, he obtains (a) a tetrahedron, (fi) an octahedron, (y) an icosahedron 
respectively. For the fourth element (earth), four isosceles right-angled triangles 
are first put together so as to form a square, and then six of these squares are 
put together to form a cube. Now, says Aristotle (299 b 23), '*it is absurd that 
planes should only admit of being put together so as to touch in a line', for just 
as a line and a line are put together in both ways, lengthwise and breadthwise, r 
so must a plane and a plane. A line can be combined with a line in the sense 
of being a line superposed^ and not adde(P^\ the inference being that deplane can | 
be superposed on biplane. Now this is precisely the sort of motion in question ! 
here; and Aristotle, so far from denying its permissibility, seems to blame ^ 
Plato for not using it. Cf. also Physics v. 4, 228 b 25, where Aristotle speaks ' 
of *' the spiral or other magnitude in which any part will not coincide with , 
any other part," and where superposition is obviously contemplated. 

Motion without deformation. 

It is well known that Helmholtz maintained that geometry requires us to 
assume the actual existence of rigid bodies and their free mobility in space, . 
whence he inferred that geometry is dependent on mechanics. 

Veronese exposed the fallacy in this (Fondamenti di geometria^ pp. xxxv — 
^Dovi, 239 — 240 note, 615 — 7), his argument being as follows. . Since geometry 
is concerned with empty space, which is immovable, it would be at least strange 
if it was necessary to have recourse to the real motion of bodies for a definition. 



i 



I UC.M4] NOTE ON COMMON NOTION 4 227 

and for the proof of the properties, of immovable space. We must distinguish 
\ the intuitive principle of motion in itself from that of motion without deforma- 
tion. Every point of a figure which moves is transferred to another point in 
space. " Without deformation " means that the mutual relations between the 
points of the figure do not change, but the relations between them and other 
figures do change (for if they did not, the figure could not move). Now 
consider what we mean by saying that, when the figure A has moved from 
the position Ax to the position ^,, the relations between the points of A in 
the position A^ are unaltered from what they were in the position A^^ are the 
same in fact as if ^ had not moved but remained at A^^ We can only say 
that, judging of the figure (or the body with its physical qualities eliminated) 
by the impressions it produces in us during its movement, the impressions 
produced in us in the two different positions (which are in tim^ distinct) 
are equal. In fact, we are making use of the notion of equality between two 
distinct figures. Thus, if we say that two bodies are equal when they 
can be superposed by means of movement without deformation^ we are com- 
mitting di petitio principii. The notion of the equality of spaces is really prior 
to that of rigid bodies or of motion without deformatioa Helmholtz supported 
his view by reference to the process of measurement in which the measure 
must be, at least approximately, a rigid body, but the exbtence of a rigid body 
as a standard to measure by, and the question how we discover two equal 
spaces to be equal, are matters of no concern to the geometer. The method 
of superposition, depending on motion without deformation, is only of use as 
9i practical test \ it has nothing to do with the theory of geometry. 

Compare an acute observation of Schopenhauer (Die Welt als WilU^ 2 ed. 
1844^ II. p. 130) which was a criticism in advance of Helmholtz' theory : "I 
am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the 
eighth is not rather attacked : ' Figures which coincide (sich decken) are 
equal to one another.' For coincidence (das Sichdecken) is either mere 
tautology, or something entirely empirical, which belongs, not to pure intuition 
(Anschauung), but to external sensuous experience. It presupposes in fact 
the mobility of figures; but that which is movable in space is matter and 
nothing else. Thus this appeal to coincidence means leaving pure space, the 
sole element of geometry, in order to pass over to the material and empirical." 

Mr Bertrand Russell observes (En^clopaedia BritannicOy Suppl. Vol. 4, 
1902, Art '* Geometry, non-Euclidean ") that the apparent use of motion here 
is deceptive ; what in geometry is called a motion is merely the transference 
of our attention from one figure to another. Actual superposition, which is 
nominally employed by Euclid, is not required; all that is required is the 
transference of our attention from the original figure to a new one defined by 
the position of some of its elements and by certain properties which it shares 
with the original figure. 

If the method of superposition is given up as a means of defining theoreti- 
cally the equab'ty of two figures, some other definition of equality is necessary. 
But such a definition can be evolved out of empirical or practical observation 
of the result of superposing two material representations of figures. This is 
done by Veronese (Ekmenti di geometria^ 1904) and Ingrami (Elementi di 
geometria, 1904). Ingrami says, namely (p. 66): 

" If a sheet of paper be folded double, and a triangle be drawn upon it 
and then cut out, we obtain two triangles superposed which we in practice call 
equal. If points ^, J?, C, Z> ... be marked on one of the triangles, then, 
when we place this triangle upon the other (so as to coincide with it), we see 

IS— 2 



228 BOOK I [l C. N. 4 

that each of the particular points taken on the first is superposed on one 
particular point of the second in such a way that the segments AB^ AC, AD, 
BQ BD^ CDy ... are respectively superix>sed on as many segments in die 
second triangle and are therefore equal to them respectively. In this way we 
justify the following 

'' Definition of equality. 

''Any two figures whatever will be called equal when to the pdnts of one 
the points of the other can be made to correspond univacalh [Le. every 0me 
point in one to one distinct point in the other and viu vena] in such a way 
that the segments which join the points, two and two, in one figure are 
respectively equal to the segments which join, two and two, the corresponding 
pomts in the other." 

Ingrami has of course previously postulated as known the signification of 
the phrase equa/ {rectilineal) segments, of which we get 9i practical notion when 
we can place one upon the other or can place a third movable segment 
successively on both. 

New systems of Congruence-Postulates. 

In the third Article of Questiom riguardanti la geametria ekmentart, ipop, 
pp. 65 — 82, a review is given of three difierent systems: U) that of Pas(£ in 
Vorl^ngen Oder neuere Geamdrk, 1882, p. loi sqq., (2) that of Veronese 
according to the FondammH H geametria, 1891, md die ElemenH taken 
together, (3) that of Hilbert (see Chundlagen der Geametrie, 1903, pp. 7 — 15). 

These systems difier in the particular conceptions taken by the three 
authors as primary, (i) Pasch considers as primary the notion of congruence 
or equality between any figures which are made up of a finite number of points 
only. The definitions of congruent segments and of congruent angles have to 
be deduced in the way shown on pp. 68 — 9 of the Article refened to, after 
which EucL i. 4 follows immediately, and Eucl. i. 26 (i) and i. 8 by a 
method recalling that in Eucl. i. 7, 8. 

(2) Veronese takes as primary the conception of congruence between 
segments (rectilineal). The transition to congruent angles^ and thence to 
triangles is made by means of the following postulate : 

''Let AB^ AC and A'B\ AC be two pairs of straight lines intersecting 
at A^ A\ and let there be determined upon them the congruent segments 
AB, AB* and the congruent segments AC, AC\ 

then, if BC^ B*C are congruent, the two pairs of straight Una are con- 
gruent" 

(3) Hilbert takes as primary the notions of congruence between twth 
segments and angles, . 

It is observed in the Article referred to that, from the theoretical stand- [ 
point, Veronese's system is an advance upon that of Pasch, since the idea of 
congruence between segments is more simple than that of congruence between t 
any figures ; but, didactically, the development of the theory is more compli- ' 
cated when we start from Veronese's system than when we start from that of \ 
Pasch. 

The system of Hilbert offers advantages over both the others from the 
point of view of the teaching of geometry, and I shall therefore give a short 
account of his system only, following the Article above quoted. 



i 



I. C, N. 4] NOTE ON COMMON NOTION 4 229 

Hubert's system. 

The following are substantially the Postulates laid down. 

(i) If one s^mmt is congruent ivith another^ the second is also congruent 
with the first, 

(2) If an angle is congruent with another angU^ the second angle is also 
congruent with the first, 

(3) Two segments congruent with a third are congruent with one another, 

(4) 7\vo angles congruent with a third are congruent with one another, 

(5) Any segment AB is congruent with itself^ independently of its sense. 
This we may express symbolically thus : 

AB^AB^BA, 

(6) Any angle (ab) is congruent with itself independently of its sense. 
This we may express symbolically thus : 

(ab) = {ab) = (ba), 

(7) On any straight line r*, starting from any one of its points A', and on 
each side of it respectively^ there exists one and only one segment congruent with a 
segmeftt AB belonging to the straight line r. 

(8) Given a ray a, issuing from a point O, in any plane which contains it 
and on each of the two sides of it^ there exists one and only one ray b issuing 
from O stich that the angle (ab) is congruent with a given angle (a'b'). 

(9) ^ AB, BC are two consecutive segments of the same straight line r 
{segments^ that is, having an extremity and no other point common), and A'B', 
B'C two consecutive segments on another straight line r', and if AB = A'B', 
BC s B'C, then 

AC^AC, 

I (10) j^ (ab), (be) are two consecutiife angles in the same plane ir (angles, 

(that is, having the ifertex and one side common), and (a'b'), (b'c') two consecu- 
tive angles in another plane v, and if (A) = (a'b'), (be) = (b'c'), then 

^ (11) If two trian^s have two sides and the included angles respectively 

congruent, they have also their third sides congruent as well as the angles 
opposite to the congruent sides respectiwly, 

I As a matter of fact, Hilbert's postulate corresponding to ^11) does not 



( 



assert the equality of the third sides in each, but only the equality of the two 
remaining angles in one triangle to the two remaining angles in the other 
respectively. He proves the equality of the third sides (thereby completing 
the theorem of Eucl. i. 4) by reductio 

ad absurdum thus. ljtiABC,AB'C' A A' 

be the two triangles which have the ^^ ^y^^ 

sides AB, AC respectively congruent ^/^ \ ^/^ A 

with the sides AB', A'C and the ^^ \ y^ \ \ 

included angle at A congruent with ^ ^ ^7- g-jj. 

the included anple at A\ 

Then, by Hilbert's own postulate, the angles ABC, ABC are congruent, 
as also the angles ACB, AC'B, 

If BC is not congruent with BC, let D be taken on BC such that BC, 
BD are congruent, and join AD, 




7 






930 BOOK I [i. C.N.A 

Then the two triangles ABQ ASD have two rides and the included 
angles congraent respectively; therefore, by the same postulate^ the angles 
BACy BAD are congruent 

But the angles BAC^ BA'C are congruent ; therefore, by (4) above, the 
angles BAC^ BAD are congruent : which is impossible, since it contradicts 
(8) above. 

Hence BC^ BC cannot but be congruent 

Eucl. I. 4 is thus proved \ but it seems to be as well to indude all of that 
theorem in the postulate, as b done in ^i i) above, rince the two parts of it are 
equally suggested by empirical observation of the result of one superporition. 

A proof similar to that just given immediately establishes Eud. i. 26 (i), 
and Hilbert next proves that 

If two angles ABC, A'B'C are eangnteni with one another^ their supplt- 
mentary angles CBD, CB'D' are also eo^pnent with one another. 

We choose Ay D on one of the straight lines forming the first angle, and 
A\ D on one of those forming the second angle, and agam C, C on the other 

O C 

/^^^^^ 

straight lines forming the angles, so that AB is congruent with AB^ C*B 
with CB, and DB with DB. 

The triangles ABC, ABC are congruent, by (11) above; and AC is 
congruent with AC, and the angle CAB with the angle CAB. 

Thus, AD, AD being congruent, by (9), the triangles CAD, CAB are 
also congruent by (11); / 

whence CD is congruent with CB, and the angle ADC with the angle 
ADC. 

Lastly, by (11), the triangles CDB, CDB are congruent, and the angles 
CBD, CBD are thus congruent 

Hilbert's next proposition is that | 

Given that the angle (h, k) in the plane a is congruent with the angle (h', k') I 
in the plane a, and that 1 is a half-ray in the plane a starting from the vertex | 
of the angle (h, k) and lying within that angle, there always exists a half-ray V [ 
in the second plane a, starting from the vertejC of the an^ (h', k') and lying i 
within that an^ such that y 

(h,i)=(h'.i'). ««/(k.i)=(k',i'). • 

If O, O are the vertices, we choose points A, B on h, h, and points A, B t 
on K, K respectively, such that OA, OA are congruent and also OB, OB: * 

\ 

( 



The triangles OAB, OAB are then congruent ; and, if / meets AB in C, 
we can determine C on AB such that ^'C is congruent with AC. 
Then f drawn from O through C is the half-ray required. 



T" 



I. C N. 4] 



NOTE ON COMMON NOTION 4 



a3i 



The congruence of the angles (^ /), {h\ f) follows from (11) directly, and 
that of (k, i) and (^, f) follows in the same way after we have inferred by 
means of (9) that, AB, AC being respectively congruent with A'F, A*C\ the 
difference BC is congruent with the difference ffC\ 

It is by means of the two propositions just given that Hilbert proves that 
All right angles are congruent with one another. 

Let the angle BAD be congruent with its adjacent angle CAD^ and 
Ukewise the angle BAU congruent with its adjacent angle CA'Jff. All four 
angles are then right angles. 



O'T 



QW 



D' 



If the angle BA'D is not congruent with the angle BAD, let the angle 
with AB for one side and congruent with the angle BA*D be the angle 
BAD\ so that AD' falls either within the angle BAD or within the angle 
DAC. Suppose the former. 

By the last proposition but one (about adjacent angles), the angles 
BA D^ BAD* being congruent, the angles CA*D, CAD' are congruent 

Hence, by the hypothesis and postulate (4) above, the angles BAD\ 
CAD* are also congruent 

And, since the angles BAD, CAD are congruent, we can find within the 
angle CAD a half-ray CAD'* such that the angles BAD\ CAD'' are 
congruent, and likewise the angles DAD\ DAD" (by the last proposition). 

But the angles BAD\ CAD' were congruent (see above); and it 
follows, by (4), that the angles CAD'^ CAD" are congruent: which is 
impossible, since it contradicts postulate (8). 

Therefore etc 

Euclid I. 5 follows directly by applying the postulate (11) above to ABC, 
A CB as distinct triangles. 

Postulates (9), (10) above give in substance the proposition that "the 
sums or differences of s^;ments, or of 
angles, respectively equal, are eqxiaL" 

Lastly, Hilbert proves EucL i. 8 by 
means of the theorem of Eud. i. 5 and 
the proposition just stated as applied to 
angles. 

ABC, A'ffC being the given triangles 
with three sides respectively congruent, 
we suppose an angle CBA" to be deter- 
mined, on the side of BC opposite to A, 
congruent with the angle A'BC, and we make BA" equal to A'B, 

The proof is obvious, being equivalent to the alternative proof often given 
in our text-books for Eud. i. 8. 




232 BOOK I [i. CN.S 

Common Notion 5. 

fcoi ri i\o¥ rw fiifHw^ fi€l(fi¥ [Ivrtr]. 
7^ whole is greater than the pari. 

Proclus includes this "axiom " on the same ground as the preceding one. 
I think however there is force in the objection which Tannery takes to it, 
namely that it replaces a difertni expression in EudL i. 6, where it is stated 
that "the triangle DBC wdl be equal to the triangle ACB^ the ks$ to the 
greater: which is absurd!* The axiom appears to be an abstraction or 
generalisation substituted for an immediate inference from a geometrical 
figure, but it takes the form of a sort of definition of whole and part The 
probabilities seem to be against its being genuine, notwithstanding Proclus* 
approval of it 

Qavius added the axiom that the whok is the equal to the sum of its parts. 

Other Axioms introduced after Euclid's time. 

[9] Ikuo straight lines do not enclose (or eoniain) a space. 

Proclus (g. 196, 21) mentions this in illustration of the undue multiplication 
of axioms, and he points out, as an objection to it, that it belongs to the 
subject matter of geometry, whereas axioms are of a general character, and 
not peculiar to any one science. The real objection to the axiom is that it is 
unnecessary, since the fact which it states is included in the meanina; of 
Postulate I. It was no doubt taken firom the passage in i. 4, ''if...the oase 
BC does not coincide with EF^ two straight lines wul enclose a space-, which 
is impossible** \ and we must certainly re^ud it as an interpolation, notwith- 
standing that two of the best iiss. have it after Postulate 5, and one gives it 
as Common Notion 9. 

Pappus added some others which Proclus objects to (p. 198, 5) because 
they are either anticipated in the definitions or follow firom them. 

te) ' All the parts of a plane^ or of a straight line, coincide with one another. 

(h) A point divides a line^ a line a surface, and a surf cue a solid \ on which 
Proclus remarks that everything is divided by the same things as those by 
which it is bounded. 

An-Nairizi (ed. Besthom-Heibcrg, p. 31, ed. Curtze, p. 38) in his version 
of this axiom, which he also attributes to Pappus, omits the reference to 
solids, but mentions planes as a particular case of surfaces. 

" (a^ A surface cuts a surface in a line ; 
(P) If two surfaces which cut one another are plane, they cut one another 

in a straight line ; 
(y) A line cuts a line in a point (this last we need in the first proposition)." ^ 

{h) Magnitudes are susceptible of the infinite (or unlimited) both by way of 
addition and by way of successive diminution, but in both cases potentially only [ 
M airttfioy iv rocs iirfiOtfriv iartv jcou rg irpo<rO€(r€i jccu rg iinKa$<up4a'€L, Swofici 
Sk heirtpov). i 

An-Nairizl's version of this refers to straight lines and plane surfaces only : ' 
"at regards the straight line and the plane surface, in consequence of their . 
evenness, it is possible to produce them indefinitely. 

This "axiom" of Pappus, as quoted by Proclus, seems to be taken directly I 
firom the discussion of ro an-cipov in Aristotle, Physia in. 5 — 8, even to the 
wording, for, while Aristotle uses the term division (8ia(Jpc(ric) most frequently 
as the antithesis of addition (<rw0c(ricV he occasionally speaks of subtraction 
(d^aipco-if) and diminution (Ka^oipco-ic). Hankel {Zur Geschichte der Mathe- 
matih im Alterthum und MittelaUer, 1874, pp. 119 — 120) gave an admirable 



f 



\ 



1. Axx.] ADDITIONAL AXIOMS 233 

summary of Aristotle's views on this subject ; and they are stated in greater 
detail in Gorland, AristoteUs und die Mathematik^ Marburg, 1899, PP* ^57 — 
183. The infinite or unlimited (dirc4f>ov) only exists potentially (8wa/ict), not 
in actuality (^cpycti^). The infinite is so in virtue of its endlessly changing 
into something else, like day or the Olympic Games {Phys, in. 6, 206 a 15 — 25). 
The infinite is manifested in different forms in time, in Man, and in the 
division of magnitudes. For, in general, the infinite consists in something new 
being continually taken, that something being itself always finite but always 
different. Therefore the infinite must not be regarded as a particular thing 
(to3c ri), as man, house, but as being always in course of becoming or decay, 
and, though finite at any moment, always different from moment to moment. 
But there is the distinction between the forms above referred to that, whereas 
in the case of magnitudes what is once taken remains, in the case of time and 
Man it passes or is destroyed but the succession is unbroken. The case of 
addition is in a sense the same as that of division ; in the finite magnitude the 
former takes place in the converse way to the latter ; for, as we see the finite 
magnitude divided ad infinitum^ so we shall find that addition gives a sum 
tending to a definite limit I mean that, in the case of a finite magnitude, 
you may take a definite fraction of it and add to it (continually) in the same 
ratio ; if now the successive added terms do not include one and the same 
magnitude whatever it is [i.e. if the successive terms diminish in geometrical 
progression], you will not come to the end of the finite magnitude, but, if the 
ratio is increased so that each term does include one and the same magnitude 
whatever it is, you will come to the end of the finite magnitude, for every 
finite magnitude is exhausted by continuaUy taking from it any definite 
fraction whatever. Thus in no other sense does the infinite exist, but only 
in the sense just mentioned, that is, potentially and by way of diminution 
(206 a 25 — b 13). And in this sense you may have potentially infinite 
addition^ the process being, as we say, in a manner, the same as with division 
ad infinitum : for in the case of addition you will always be able to find some- 
thing outside the total for the time being, but the total will never exceed every 
definite (or assigned) magnitude in the way that, in the direction of division, 
the result will pass every definite magnitude, that is, by becoming smaller 
than it The infinite therefore cannot exist even potentially in the sense of 
exceeding every finite magnitude as the result of successive addition (206 b 
16 — 22). It follows that the correct view of the infinite is the opposite of 
that commonly held : it is not that which has nothing outside it, but that 
which always has something outside it (206 b 33 — 207 a i). 

Contrasting the case of number and magnitude, Aristotle points out that 
(i) in number there is a limit in the direction of smallness, namely unity, but 
none in the other direction : a number may exceed any assigned number 
however great ; but (2) with magnitude the contrary is the case : you can find 
a magnitude smaller than any assigned magnitude, but in the other direction 
there is no such thing as an infinite magnitude (207 b r-^5). The latter 
assertion he justified by the following argument However large a thin^ can 
be potentially, it can be as large actually. But there is no magnitude 
perceptible to sense that is infinite. Therefore excess over every assigned 
magmtude is an impossibility; otherwise there would be something lu^er 
than the universe (o^yos) (207 b 17 — 21). 

Aristotle is aware that it is essentially of physical magnitudes that he is 
speaking. He had observed in an earlier passage {Phys, in. 5, 204 a 34) that 
it is peiiiaps a more general inquiry that would be necessary to determine 



I 



234 BOOK I [i. Axx. 

whether the infinite is possible in mathematics, and in the domain of thought 
and of things which have no magnitude; but he excuses himself from entenng 
upon this inquiry on the ground that his subject is physics and sensible 
objects. He returns however to the bearing of his conclusions on mathematics 
in III. 7, 207 b 27 : "my argument does not even rob mathematicians of their 
study, although it denies the existence of the infinite in the sense of actual 
existence as something increased to such an extent that it cannot be gone 
through (dSic^tryrov) ; for, as it is, thc^ do not even need the infinite or use 
it, but only require that the finite (straight line) shall be as long as they please; 
and another magnitude of any size whatever can be cut in the same ratio as 
the greatest magnitude. Hence it will make no difference to them for the 
purpose of demonstration." 

Lastly, if it should be urged that the infinite exists in thought^ Aristotle 
replies that this does not involve its existence in fact, A thing is not greater 
than a certain size because it is conceived to be so, but because it i>; and 
magnitude is not infinite in virtue of increase in thought (208 a 16 — 22). 

Hankel and Gorland do not quote the passage about an infinite series of 
magnitudes (206 b 3 — 13) included in the above paraphrase; but I have 
thought that mathematicians would be interested in the distinct e3q>ression of 
Aristotle's view that the existence of an infinite series the terms of which are 
magnitudes is impossible unless it is convergent, and (with reference to 
Riemann's developments) in the statement that it does not matter to geometiy 
if the straight line is not infinite in length, provided that it is as long as we 
please. 

Aristotle's denial of even the potential existence of a sum of magnitudes 
which shall exceed every definite magnitude was, as he himself implies, in 
conflict with the lemma or assumption used by Eudoxus (as we infer firom 
Archimedes) to prove the theorem about the volume of a pyramid. The 
lemma is thus stated by Archimedes (Quadrature of a parabola^ preface): 
" The excess by which the greater of two unequal areas exceeds the less can, 
if it be continually added to itself, be made to exceed any assigned finite 
area." We can therefore well understand why, a century later, Archimedes 
felt it necessary to justify his own use of the lemma as he does in the same 
preface : " The earlier geometers too have used this lemma : for it is by its 
help that they have proved that circles have to one another the duplicate 
ratio of their diameters, that spheres have to one another the triplicate ratio 
of their diameters, and so on. And, in the result, each of the said theorems 
has been accepted no less than those proved without the aid of this lemma." 

Principle of continuity. 

The use of actual construction as a method of proving the existence of J 
figures having certain properties is one of the characteristics of the Elements. 
Now constructions are effected by means of straight lines and circles drawn I 
in accordance with Postulates i — 3 ; the essence of them is that such straight ^ 
lines and circles determine by their intersections other points in addition to \ 
those given, and these points again are used to determine new lines, and so on. 
This being so, the existence of such points of intersection must be postulated ) 
or proved in iht same way as that of the lines which determine them. Yet ] 
there is no postulate of this character expressed in Euclid except Post. 5. / 
This postulate asserts that two straight lines meet if they satisfy a certain 
condition. The condition is of the nature of a &opiirfio« (discrimination^ or 
condition of possibility) in a problem ; and, if the existence of the point of 



r 



I.AXX.] PRINCIPLE OF CONTINUITY 235 

intersection were not granted, the solutions of problems in which the points of 
intersection of straight lines are used would not in general furnish the required 
proofs of the existence of the figures to be constructed. 

But, equally with the intersections of straight lines, the intersections of 
circle with straight line, and of circle with circle, are used in constructions. 
Hence, in addition to Postulate 5, we require postulates asserting the actual 
existence of points of intersection of circle with straight line and of circle 
with circle. In the very first proposition the vertex of the required equilateral 
triangle is determined as one of the intersections of two circles, and we need 
therefore to be assured that the circles will intersect Euc lid seems to assume 
U ita s obvious, although it is not so ; and he makes a similar assumption m 
XTii, It is true that in the latter case Euclid adds to the enunciation that 
two of the given straight lines must be together greater than the third ; but 
there is nothing to show that, if this condition is satisfied, the construction is 
always possible. In i. 12, in order to be sure that the circle with a given 
centre will intersect a given straight line, Euclid makes the circle pass through 
a point on the side of the line opposite to that where the centre is. It appears 
therefore as if, in this case, he based his inference in some way upon the 
, definition of a circle combined with the fact that the point within it called 
the centre is on one side of the straight line and one point of the circumference 
on the other, and, in the case of two intersecting circles, upon similar con- 
siderations. But not even in Book in., where there are several propositions 
about the relative positions of two circles, do we find any discussion of the 
conditions under which two circles have two, one, or no point common. 

The deficiency can only be made good by the Principle of Continuity, 

Killing {Einfiihrung in die Grundi^gen der Geometrie^ 11. p. 43) gives the 
following forms as sufficient for most purposes. 

(a) Suppose a line belongs entirely to a figure which is divided into two 
parts ; then, if the line has at least one point common with each part, it must 
also meet the boundary between the parts; or 

(p) If a point moves in a figure which is divided into two parts, and if it 
belongs at the beginning of the motion to one part and at the end of the 
motion to the other part, it must during the motion arrive at the boundary 
between the two parts. 

In the Questioni riguardanti la geometria elementare^ Article 4 (pp. 83 — loi), 
the principle of continuity is discussed with special reference to the Postulate 
of Dedekmd, and it is shown, first, how the Postulate may be led up to and, 
secondly, how it may be applied for the purposes of elementary geometry. 

Suppose that in a segment AB of a straight line a point C determines 
two segments A C, CB, If we consider the point C as belonging to only one 
of the two segments ACy CB^ we have a division of the segment AB into 
two parts with the following properties. 

1. Every point of the segment AB belongs to one of the two parts. 

2. The point A belongs to one of the two parts (which we will call the 
firsi) and the point B to the other ; the point C may belong indifferently to 
one or the other of the two parts according as we choose to premise. 

3. Every point of the first part precedes every point of the second in the 
order AB of the segment 

(For generality we may also suppose the case in which the point C fedls at 
A or at B. Considering C, in these cases respectively, as belonging to the 
first or the second part, we still have a division into parts which have the 
properties above enunciated, one part being then a single point A or B.) 



L. 



236 BOOK I [lAxx. 



Now, considering carefully the inverse of the above proposition, we see 
that it agrees with the idea which we have of the continuity of the straight 
line. Consequently we are induced to admit as 9,poshdaU the following. 

If a segment of a straight line AB is divided into two parts so that 
{iS every point of the segment AB bdongs to one of theparts^ 

(2) t?u extremity A belongs to the first part and "^ to the second^ and 

(3) any point whatever of the first part precedes any point whatever ef ike 
seamd party in the order AB of the segment^ 

there exists a point C of the segment AB (which may belong either to ame 
part or to the other) such that every point of AB that precedes C belongs to the 
first party and every point of AB that follows C belongs to the second part in 
the division originally assumed 

(If one of the two parts consists of the single point A or B^ the point C 
is the said extremity A ox B oi the segment) 

This is the Postulate of Dedekind, which was enunciated by Dedekind 
himself in the following slightly different form (Stetigheit und irrationale ZcMen^ 
1872, new edition 1905, p. 11). 

'' If all points of a straight line fall into two classes such that every paint of 
the first class lies to the left of every point of the second class^ there exists one emd 
only one point which produces this division of all the points into two classes^ this 
division of the straight line into two parts,** 

The above enunciation may be said to correspond to the intuitive notion 
which we have that, if in a segment of a straight line two points start from 
the ends and describe the segment in opposite senses, they meet in a point 
The point of meeting might be regarded as belonging to both parts, but for 
the present purpose we must regard it as belonging to one only and subtracted 
from the other part 

Application of DedehincPs postulate to angles. 

If we consider an angle less than two right angles bounded by two rays 
a, by and draw the straight line connecting Ay a point on a, with By a point 4 
on by we see that all points on the finite s^ment AB correspond univocally to 
all the rays of the angle, the point corresponding to any ray being the point 
in which the ray cuts the segment AB \ and if a ray be supposed to move 
about the vertex of the angle from the position a to the position by the 
corresponding points of the segment AB are seen to follow in the same 
order as the corresponding rays of the angle {ab). 

Consequently, if the angle (ab) is divided into two parts so that 

(i) each ray of the angle (ab) belongs to one of the two parts, 

it) the outside ray a belongs to the first part and the ray b to the second, 

(3) any ray whatever of the first part precedes any ray whatever of the ] 

second part, i 

the correspondmg points of the segment AB determine two parts of the i 

segments such that \ 

ii) every point of the segment AB belongs to one of the two parts, ) 

2^ the extremity A belongs to the first part and B to the second, 
3) any point whatever of the first part precedes any point whatever of '; 
the second. I 

But in that case there exists a point C of AB (which may belong to one 
or the other of the two parts) such that every point of AB that precedes C 
belongs to the first part and every point of AB that follows C belongs to the 
second part 






I 



s 



I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 237 

Thus exactly the same thing holds of Cy the ray corresponding to C, with 
reference to the division of the angle (ab) into two parts. 

It is not difficult to extend this to an angle (ab) which is either flat or 
greater than two right angles ; this is done (Bonola, op, cit. pp. 87 — 88) by 
supposing the angle to be divided into two, (ad)y {db)^ each less than two 
ri^t angles, and considering the three cases in which 

(i) Sie ray ^ is such that all the rays that precede it belong to the first 
part and those which follow it to the second part, 
the ray ^ is followed by some rays of the first part, 
the ray d is preceded by some rays of the second part 

Application to circular arcs. 

If we consider an arc AB of a circle with centre Oy the points of the arc 
correspond univocally, and in the same order, to the rays from the point O 
passing through those points respectively, and the same argument by which 
we pa^ed from a segment of a straight line to an angle can be used to make 
the transition from an angle to an arc. 

Intersections of a straight line with a circle. 
It is possible to use the Postulate of Dedekind to prove that 

If a straight line has one point inside and one point outside a circle^ it has 
two points common with the circle. 

For this purpose it is necessary to assume (i) the proposition with reference 
to the perpendicular and obliques drawn from a given point to a given straight 
line, namely that of all straight lines drawn from a given point to a given 
straight line the perpendicular is the shortest, and of the rest (the obliques) 
that is the longer which has the longer projection upon the straight line, while 
those are eqiud the projections of which are equal, so that for any given 
length of projection there are two equal oblic^ues and two only, one on each 
side of the perpendicular, and (2) the proposition that any side of a triangle 
is less than the sum of the other two. 

Consider the circle (C) with centre Oy and a straight line (r) with one 
point A inside and one point B outside the 
circle. 
) By the definition of the circle, if J? is /^ p AJJK B 

the radius, 

OA<Ry OB>R. 

Draw OP perpendicular to the straight 
line r. 

Then 0P< OAy so that OP is always 
less than J?, and P is therefore within the 
circle C 

Now let us fix our attention on the finite segment AB of the straight 
line r. It can be divided into two parts, (i) that containing all the points H 
for which OH<R (i.e. points inside C), and (2) that containing all the 
points K for which OK ^ R (points outside C or on the circumference of C). 
m Thus, remembering that, of two obliques from a given point to a given 
straight line, that is greater the projection of which is greater, we can assert 
that all the points of the segment PB which precede a point inside C are 
inside C, and those ^iYa^ follow a point on the circumference of C or outside 
t C are outside C. 

Hence, by the Postulate of Dedekind, there exists on the segment PB a 




f 



238 ROOK I [1. Axx. 

point Jf such that all the points which precede it belong to the fint part and 
those which follow it to the second part 

I say that Afis common to the straight line rand the circle C, or 

For suppose, e.g., that OJIf< R. 

There will then exist a segment (or length) <r less than the difference 
between R and OM. 

Consider the point M*^ one of those which foUcw M^ such that MMT is 
equal to <r. 

Then, because any side of a triangle is less than the sum of the other two, 
OM' < OM^ MMT. 

But OM^MM'^OM-¥ir<X, 

whence OM'<X^ 

which is absurd. 

A similar absurdity would follow if we suppose that OM > X. 

Therefore OM must be equal to R. 

It is immediately obvious that, corresponding to the point if on the segment 
F£ which is common to r and C, there is another pomt on r which. has the 
same property, namely that which is symoaetrical to AT with respect to P. 

And Uie proposition is proved 

Intersections of two circles. 

We can likewise use the Postulate of Dedekind to prove that 

If in a givtn plane a drde C has onepoint X inside and^nepaini Y auisUe 
another drde C, the two cirdes intersect in two points. 

We must first prove the following 

Lemma, 

If Oj O' are the centres of two circles (7, C\ and J?, R' their radii 
respectively, the straight line 00' meets the circle C in two points A^ B^ on6 
of which is inside C and the other outside it ^ 

Now one of these points must fall (i) on the prolongation of ffO 
beyond O or {2) on Off itself or (3) on the ( 

prolongation of Off beyond ff. 

(i) First, suppose A to lie on ffO pro- 
duced. 

Then Aff^AO-^OO'-^R^ Off (a). 

But, in the triangle Off V, 

ffY<OY^Off, 
and, since ffY>R, OY^R, 
R<R^Off. 

It follows from M that Aff>R\ and A 
therefore lies outside C, 

(2) Secondly, suppose A to lie on 
Off. 

Then Off ^OA^Aff^R^Aff ,,.{fi). 
From the triangle OffX we have b 

Off^OX^ffX, 
and, since OX^R^ ffX<R^ it follows 
that 

Off<R^R, 
whence, by {fi\ Aff <R,so that A lies inside C. 





I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 239 

(3) Thirdly, suppose A to He on OO produced. 

Then R^OA^OO^OA (y). 

And, in the triangle OOX^ 

OX < 00^ OX, Y/ 

thatb R<.0O^^(yX, ^\ 

whence, by (y), 

oo^o'A<oa^ox, c\ 

or O'A < ax, 

and A lies inside C 

It is to be observed that one of the two points A, B is in the position of 
case (i) and the other in the position of either case (2) or case (3) : whence 
we must conclude that one of the two points ^, ^ is inside and the other 
outside the circle C\ 

Proof of theorem. 

The circle C is divided by the points A, B into two semicircles. Consider 
one of them, and suppose it to be 
described by a point moving from A 
to B.^ 

Take two separate points F, Q 
on it and, to fix our ideas, suppose 
that P precedes Q. 

Comparing the triangles OOP, 
OOQymt observe that one side OO 
is common, OP is equal to OQ, and 
the angle POO is less than the angle 
QOO. 

Therefore OP<OQ. 

Now, considering the semicircle APQB as divided into two parts, so that 
the points of the first part are inside the circle C\ and those of the second 
part on the circumference of C or outside it, we have the conditions necessary 
for the applicability of the Postulate of Dedekind (which is true for arcs of 
circles as for straight lines) ; whence there exists a point M separating ttu two 
parts, 
' I say that OM^ R. 

\ For, if not, suppose OM < R!, 

ilf then o- signifies the difference between R and OM, suppose a point M\ 
which follows M, taken on the semicircle such that the chord MM' is not 
greater than o- (for a way of doing this see below). 
^ Then, in the triangle OMM*, 

OM* < OM^ MM* < OM-¥ <r, 
and therefore OM' < R. 

It follows that AT, a point on the arc MB, is inside the circle C : 
which is absurd. 

Similarly it may be proved that OMis not greater than R. 
Hence OM=R. 

[To find a point M such that the chord MM* is not greater than <r, we 
may proceed thus. 

Draw from Mb, straight line MP distinct from OM, and cut off MP on it 
equal to v/i. 




240 



BOOK I 



[i. Axx. 



Join OP^ and draw another radius OQ such that the ang^e POQ is equal 
to the angle MOP. q 

The intersection, M\ of OQ with the 
circle satisfies the required condition. 

For MM' meets OP at right angles 
in 5. 

Therefore, in the right-ai^led triangle 
MSP^ MS is not greater than MP (it is 
less, unless MP coincides with if5, when 
it is equal). 

Therefore MS is not greater than <r/2, so that MMT is not greater than v.] 




J 



i i 

I 

I 

!( 

ft ■ 



B(X)K I. PROPOSITIONS. 

Proposition i. 

I On a given finite straight line to construct an equilateral 

triangle. 

I' Let AB be the given finite straight line. 

* Thus it is required to con- 

fS struct an equilateral triangle on 
' the straight line A£. 

With centre A and distance 
AB let the circle BCD be 
described ; [Post 3] 

'^o again, with centre B and dis- 
1 tance BA let the circle ACE 
I be described ; [Post 3] 

:. and from the point C, in which the circles cut one another, to 
\ the points -/4, B let the straight lines CA^ CB be joined. 
\ [Post. I] 

15 Now, since the point A is the centre of the circle CDB^ 
j AC is equal to AB. [Def. 15] 

Again, since the point B is the centre of the circle CA£y 
J i BC is equal to BA. [Def. 15] 

But CA was also proved equal to AB ; 

iao therefore each of the straight lines CA, CB is equal to AB. 

And things which are equal to the same thing are also 

equal to one another ; [C. N. i] 

therefore CA is also equal to CB. 

Therefore the three straight lines CA^ AB^ BC are 
as equal to one another. 

H. E. 16 




BOOK I [1.1 

Therefore the triangle ABC is equilateral ; and it has 
been constructed on the given finite straight line AB. 

(Being) what it was required to do. 

I. On a given finite straight line. The Greek usage diffen finom oars in that the 
definite article is employed in sudi a phrase as this where we have the indefinite. M rft 
do$€lfffit c^ffittf w€W€pafffUpiit, **on iJkg given finite straight line," i.e. the finite straight line 
which we choose to take. 

3. Let AB be the given finite straight line. To be strictly literal we should have to 
translate in the reverse order "let the given finite straight line be the (straight line) AB^i 
bat this order is inconvenient in other cases where there is more than one datam, e^ in the 
sttting-aut of I. 3, *'let the given point be A^ and the given straight line BC^** tihe awkward- 
ness arising from the omission of the verb in the second daose. Hence I have, for deamess' 
sake, adopted the other order throagfaoat the book. 

8. let the circle BCD be described. Two things are here to be noted, (1) the elegant 
and practically universal use of the perfect passive imperative in constructions, yrfp £ f $t » 
meaning of course "let it Aatfe heen described " or "suppose it described,** (3) the impossi- 
bility o7 expressing shortly in a translation the force of the words in their original order. 
jci^irXot ytypi^ta 6 BFA means literally "let a circle have been described, the (circle, namehr, 
which I denote bv) BCD" Similarly we have lower down *' let straight lines, (namely) the 
(straight lines) CA, CB, be joined,'* iwttfB^Stfaap MtUu td FA, r£ There aecms to be 
no practicable alternative, in En^i^ bat to translate as I have done in the teat. 

13. from the point C... Eodid is carefid to adhere to the phraseoloMpr of Pottalate i 
except that he speaks of "joining" (^rc^r^dMvar) instead of " drawing '^(W^^tiv). He 
does not allow himself to use the shortenei expression " let the straight line r^ be joined " 
(without mention of the points /*, C) until i. 5. 

10. each of the straight lines CA, CB, inripa rflp TA, FB and 94. the three 
straight lines CA, AB, BC, ol rpeit al FA, AB, BF. I have, here and in all similar 
expressions, inserted the words "straight lines" which are not in the Greek. The possettion 
of the inflected definite article enables the Greek to omit the words, but tlds is not possible 
in English, and it would scarcely be Ei^lish to write "each of CA, CB" or "the three CA, 

AB, Bc:' 

It is a commonplace that Euclid has no right to assume, without pre- 
mising some postulate, that the two circles wtJl meet in a point C To 
supply what is wanted we must invoke the Principle of Continuity (see note 
thereon above, p. 235). It is sufficient for the purpose of this proposition and 
of I. 22, where there is a similar tacit assumption, to use the form of postulate 
suggested by Killing, ''/if ^ ^ne [in this case e.g^ the circumference AC£] 
belongs entirely to a figure [in this case a plane] which is divided into two parts 
[namely the part enclosed within the circumference of the circle BCD and 
the part outside that circle], and if the line has at least one point common with 
each part, it must also meet the boundary between the parts [Le. the circum- 
ference ACE must meet the circumference BCLf\,^^ 

Zeno's remark that the problem is not solved unless it is taken for granted 
that two straight lines cannot have a common s^ment has already been 
mentioned (note on Post 2, p. 196). Thus, if AC, BC meet at F before 
reaching C, and have the part FC common, the triangle obtained, namely 
FAB, will not be equilateral, but FA, FB will each be less than AB, But 
Post 2 has already laid it down that two straight lines cannot have a common 
segment 

Proclus devotes considerable space to this part of Zeno's criticism, but 
satisfies himself with the bare mention of the other part, to the effect that it 
is also necessary to assume that two circumferences (with different centres) 
cannot have a common part That is, for anything we know, there may be 
any number of points C common to the two circumferences ACE, BCD. It 
is not until in. 10 that it is proved that two circles cannot intersect in more 



< 



il 



1. 1] 



PROPOSITION I 



243 



points than two, so that we are not entitled to assume it here. The most we 
can say is that it is enough for the purpose of this proposition if one equilateral 
triangle can be found with the given base ; that the construction only gives 
two such triangles has to be left over to be proved subsequently. And indeed 
we have not long to wait ; for i. 7 clearly shows that on either side of the 
base AB only one equilateral triangle can be described. Thus i. 7 gives us 
the numlfer of solutions of which t^e present problem is susceptible, and it 
supplies the same want in i. 22 where a triangle has to be described with 
three sides of given length ; that is, i. 7 furnishes us, in both cases, with one 
of the essential parts of a complete Siapur|A<^ which includes not only the 
determination of the conditions of possibility but also the number of solutions 
(voo-axolf ^x^P<^ Proclus, p. 202, 5). This view of i. 7 as supplying an 
equivalent for iii. 10 absolutely needed in i. i and i. 22 should serve to correct 
the idea so common among writers of text-books that i. 7 is merely of use as a 
lemma to Euclid's proof of i. 8, and therefore may be left out if an alternative 
proof of that proposition is adopted. 

Agreeably to his notion that it is from i. i that we must satisfy ourselves 
that isosceles and scalene triangles actually exist, as well as equilateral triangles, 
Proclus shows how to draw, first a particular isosceles triangle, and then a 
scalene triangle, by means of the figure of the proposition. To make an 
isosceles triangle he produces AB in both directions to meet the respective 
circles in Z?, E^ and then describes 
circles with A^ B ais centres and AE^ 
BD as radii respectively. The result is 
an isosceles triangle with each of two 
sides double of the third side. To make 
an isosceles triangle in which the equal 
sides are not so related to the third side 
but have any given length would require 
. the use of i. 3 ; and there is no object in 
treating the question at all in advance of 
I. 22. An easier way of satisfying our- 
selves of the existence of some isosceles 
triangles would surely be to conceive any 
two radii of a circle drawn and their 
extremities joined. 

There is more point in Proclus* construction of a scalene triangle. Suppose 
^C to be a radius of one of the two 
circles, and D a point on AC lying in 
that portion of the circle with centre A 
which is outside the circle with centre B, 
Then, joining BD^ as in the figure, we 
have a triangle which obviously has all its 
sides unequal, that is, a scalene triangle. 

The above two constructions appear in 
an-Nairlzfs commentary under the name 
of Heron; Proclus does not mention his 
source. 

In addition to the above construction 
for a scalene triangle (producing a triangle in which the ''given" side is 
greater than one and less than the other of the two remaining sides), Heron 
has two others showing the other two possible cases, in whic^ the '' given " 
side is (i ) less than, (2) greater than, either of the other two sides. 

16 — 2 





UA 



BOOR I 
Proposition 2. 







[Postal 



To place at a given paint (as an extremity) a straight Une 
equal to a given straight line. 

Let A be the given point, and BC the given straight line. 
Thus it is required to place at the point A (as an extremity) 
5 a straight line equal to the given 
straight line BC. 

From the point A to the point B 
let the straight line AB be joined ; 

[Post i] 

and on it let the equilateral triangle 

JO DAB be constructed. [i. i] 

Let the straight lines AE, BF be 

produced in a straight line with DA^ 

DB\ [Post a] 

with centre B and distance BC let the 

15 circle CGH be described ; 

and again, with centre D and distance DG let the circle GKL 

be described. [Post 3] 

Then, since the point B is the centre of the circle CGH^ 

BC is equal to BG. 

so Again, since the point D is the centre of the circle GKL, 

DL is equal to DG. 
And in these DA is equal to DB ; 

therefore the remainder AL\s equal to the remainder 
BG. \C.N. 3] 

But BC was also proved equal to BG ; 

therefore each of the straight lines AL, BC is equal 
\oBG. 
And things which are equal to the same thing are also 
equal to one another ; [C.AI i] 

J therefore AL is also equal to BC. 

Therefore at the given point A the straight line AL is 
placed equal to the given straight line BC. 

(Being) what it was required to do. 

I. (as an extremity). I have inserted these words because **to place a stiaisht line 
at a given point '* (vp^t r^ Ba$4vri ^Mc(^)is not quite clear enough, at least in English. 

la Let the straight lines AE, BP be produced.... It will be observed ^t in this 
first application of Postulate 3, and a^ ain in i. 5, Euclid speaks of the amHnuaiiom ci the 
straight Une as that which is produced in such cases, iKfi€p\^waaif and wpoatxPtfiK^^Sttntw 
meaninglittle more than drawing straight lines " in a straight line with " the given straight 
lines. The first place in which Euclid uses phraseology exactly corresponding to onn when 



as 



I 



r 



1.2] PROPOSITION 2 24s 

speakii^ of a straight line being produced is in i. 16 :" let one tide of it, BC^ be produced 

to Z> " {wpofftKPtpX^Bw a^oO fda, wXevpii iiBT iwlrb A). 

22. the remainder AL...the remainder BO. The Greek expressions are Xocr^ ^ 
j AA and Xocrj rv BH, and the literal translation would be '*AL (or BG) remaining,** 
! but the shade of meaning conveyed by the position of the definite article can hardly be 

expressed in English. 

This proposition gives Proclus an opportunity, such as the Greek 
\ commentators revelled in, of distinguishing a multitude of ^ases. After 
I explaining that those theorems and problems are said to have cases which 

• have the same force, though admitting of a number of different figures, and 
t preserve the same method of demonstration while admitting variations of 
^ position, and that cases reveal themselves in the construction^ he proceeds to 

distinguish the cases in this problem arising from the different positions 

which the given point may occupy relatively to the given straight line. It may 

be (he says) either (i) outside the line or (2) on the line, and, if {i\ it may be 

\ either {a) on the line produced or {b) situated obliquely with regard to it ; if 

. (2), it may be either {a) one of the extremities of the line or {b) an intermediate 

> point on it It will be seen that Proclus* anxiety to subdivide leads him to 

I give a "case," (2) (a), which is useless, since in that "case" we are given 

what we are re(}uired to find, and there is really no problem to solve. As 

I Savile says, " qui quaerit ad p punctum ponere rectam aequalem r^ fiy rectae, 

j quaerit quod datum est, quod nemo faceret nisi forte insaniat" 

! Proclus gives the construction for (2) (b) following Euclid's way of taking 

t (? as the point in which the circle with centre B intersects DB produced^ and 

then proceeds to " cases," of which there are still more, which result from the 

different ways of drawing the equilateral triangle and of producing its sides. 

• This last class of " cases " he subdivides into three according as AB is 
(i) equal to, (2) greater than or (3) less than BC, Here again " case " (i) serves 
no purpose, since, if AB is equal to BC^ the problem is already solved. But 
Proclus' figures for the other two cases are worth giving, because in one of 
them the point G is on BD produced beyond Z>, and in the other it lies on 
BD itself and there is no need to produce any side of the equilateral triangle. 





A glance at these figures will show that, if they were used in the proposition, 
each of t}iem would require a slight modification in the wording (i) of the 
construction, since BD is in one case produced beyond D instead of B and 
in the other case not produced at all, (2) of the proo^ since BG, instead of 
being the difference between DG and DBy is in one case the sum of DG and 
DB and in the other the difference between DB and DG. 




346 BOOK I 

Modem editors generaUy seem to dassify the cues aooofding to Ae 
possible variations in the construction rather than according to differences in 
the data. Thus Lardner, Potts, and Todhunter distinguiui eight cases due 
to the three possible alternatives, (i) that the given point may be joined to 
either end of the given straight line, (2) that the equilateral triangle may then 
be described on either side of the joining line, and (3) that the side of the 
equilateral triangle which is produced may be produced in either direction. 
(But it should Imve been observed that, where AB is greater than BC^ tfie 
third alternative is between producing DB and not producing it at all) Potts 
adds that, when the given pomt lies either on the line or on the line pioduoed, 
the distinction which arises from joining the two ends of the line with the 
given point no longer exists, and there are only four cases of the problem 
(I think he should rather have said sohiHons). 

To distinguish a number of cases in this way was foreign to the really 
classical manner. Thus, as we shall see, Euclid's method is to give one case 
only, for choice the most difficult, leaving the reader to suf^ly the rest for 
himself. Where there was a real distinction between cases, sufficient to 
necessitate a substantial difference in the proof, the practice was to gjve 
separate enunciations and -proofs altogether, as we may see, e.g., from the 
Conies and the De sectione ratioms of Apollonius. 

Proclus alludes, in conclusion, to the error of those who proposed to solve 
I. 2 by describing a circle with the given point as centre and with a distance 
equcU to J9C, which, as he says, is a ptHtio prindpH. De Morgan puts the 
matter very clearly {Supplementary Remarks on the first six Books of EudS^s 
Elements in the Companion to the Almanac^ 1349, P- 6). We should ^'insist," 
he says, ''here upon the restrictions imposed by the first three postulat^ 
which do not allow a circle to be drawn with a compass-carried distance; 
suppose the compasses to close of themselves the moment they cease to toudi 
the paper. These two propositions [i. 2, 3] extend the power of construction 
to what it would have been if all the usual power of the compasses had been 
assumed ; they are mysterious to all who do not see that postulate iii does 
not ask for every use of the compasses P 



Proposition 3. 

Given two unequal straight lines^ to cut off front the 
greater a straight line equal to the 
less. 5. 

Let ABy C be the two given un- 
equal straight lines, and let AB be 
the greater of them. 

Thus it is required to cut off from 
AB the greater a straight line equal 
to C the less. 

At the point A let AD be placed 
equal to the straight line C ; [i. 2] 
and with centre A and distance AD let the circle DEF be 
described. [Post 3] 




*\ 



! 



1.3,4] PROPOSITIONS 2—4 247 

Now, since the point A is the centre of the circle DEF, 

AE is equal to AD. [Def. 15] 

But C is also equal to AD. 

Therefore each of the straight lines AE, C is equal to 
AD ; so that AE\s also equal to C. \C.N, i] 

Therefore, given the two straight lines AB, C, from AB 
the greater A[E has been cut off equal to C the less. 

(Being) what it was required to do. 

Proclus contrives to make a number of ''cases" out of this proposition 
also, and gives as many as eight figures. But he only produces this variety by 
practically incorporating the construction of the preceding proposition, instead 
of assuming it as we are entitled to do. If Prop. 2 is assumed, there is really 
only one " case " of the present proposition, for Potts' distinction between two 
cases according to the particular extremity of the straight line from which the 
given length has to be cut off scarcely seems to be worth making. 

Proposition 4. 

If two triangles have the two sides equal to two sides 
respectively, and have the angles contained by the equal straight 
lines equal, they will also have the base equal to the base, the 
triangle will be equal to the triangle, and the remaining angles 
5 will be equal to the remaining angles respectively, namely those 
which the equal sides subtend 

Let ABC, DEE he two triangles having the two sides 
AB, AC equal to the two sides DE, Z?-F respectively, namely 
AB to DE and AC to DE, and the angle BAC equal to the 
ID angle EDE. 

I say that the base BC is also equal to the base EE, the 
triangle ABC will be equal to the triangle DEE, and the 
remaining angles will be equal to the remaining angles 
respectively, namely those which the equal sides subtend, that 
15 is, the angle ABC to the angle DEE, and the angle ACB 
to the angle DEE. 

For, if the triangle ABC be 
applied to the triangle DEE, 
and if the point A be placed 
20 on the point D 

and the straight line AB 
on DE, 

then the point B will also coincide with E, because AB is 
equal to DE. 





348 BOOK I [1.4 

5 Again, AB coinciding with DE^ 

the straight line AC will also coincide with DF, because the 

angle BAC is equal to the angle EDF\ 

hence the point C will also coincide with the point F^ 

because AC\s again equal to DF. 
o But B also coincided with E ; 

hence the base BC will coincide with the base EF. 

[For if, when B coincides with E and C with F, the base 

BC does not coincide with the base EF, two straight lines 

will enclose a space : which is impossible. 

5 Therefore the base BC will coincide with 

EF^ and will be equal to it \C.N. 4] 

Thus the whole triangle ABC will coincide with the 
whole triangle DEF, 

and will be equal to it. 

p And the remaining angles will also coincide with the 
remaining angles and will be equal to them, 

the angle ABC to the angle DEF, 

and the angle ACB to the angle DFE. 

Therefore etc. 
^5 (Being) what it was required to prove 

I — 3. It is a fiict that Eadid*s enunciations not infreauently leave something to be 
desued in point of clearness and precision. Here he speaks of the triangles having "the 
angle equal to the angle, namely the angle contained by the equal straight lines ** (rV Ttfrfor 
ri yttwlg, Uniff fxtl '''¥ ^ f^ ^^^ t^tiQ^ vcptcxo/i^nyr), only one of the two angles being 
described in the latter expression ^n the accusative), and a similar expression in the dative 
being left to be understood of the other angle. It is curious too that, after mentioning two 
"sides" he speaks of the angles contained by the equal ** straight lines," not ^'sidesy It 
majr be that ne wished to i^ere scrupulously, at the outset, to the phraseolof^ of the 
dennitions, where the angle is the inclination to one another of two iines or strait lines. 
Similarljr in the enunciation of i. 5 he speaks of producing the equal '* straight lines " as if to 
keep stnctly to the wording of Postulate 3. 

3. respectively. I agree with Mr H. M. Taylor {Euclid, p. ix) that it is best to 
abandon the traditional translation of "each to each,' which would naturally seem to imply 
that all the four magnitudes are equal rather than (as the Greek iKaHpa ixaiip^ does) tnat 
one is eoual to one and the other to the other. 

3. tne base. Here we have the word ^au used for the first time in the Elements, 
Proclus explains it (p. 336, 13 — 15) as meaning (i), when no side of a triangle has been 
mentioned oefore, the side ** which is on a level with the sight *' (rV rp^ rg 6^ K»fi4nfw), 
and (3), when two sides have already been mentioned, the third side. Ftodus thus avoids 
the mistake made by some modem editors who explain the term exclusively with reference to 
the case where two sides have been mentioned before. That this is an error is proved (i) by 
the occurrence of the term in the enunciations of i. 37 etc. about triangles on tne same base 
and equal bases, (3) by the application of the same term to the bases of parallelograms in 
I. 35 etc. The truth is that the use of the term must have been suggested oy the practice of 
drawing the particular side horizontally, as it were, and the rest of the figure above it. The 
dose of a figure was therefore spoken of, primarily, in the same sense as ue base of anything 



■( 



J 



! 



1.4] PROPOSITION 4 249 

else, e.g. of a pedestal or column; but when, as in i. 5, two triangles were compared 
occupying other than the normal positions which gave rise to the name, and when two sides 
had been previously mentioned, the base was, as Proclus says, necessarily the third side. 

6. subtend, itwvrtlwtup ifw6, **to stretch under," with accusative. 

9. the angle B AC. The fiill Greek expression would be ^ ^6 rwv BA, AF v€pux90uhii 
YMrto, "the angle contained by the (straight lines) BjI^ AC." But it was a common practice 
of Greek geometers, e.g. of Archimedes and Apollonius (though not apparently of Euclid), to 
use the abbreviation «l BAP for «l BA, AF, **the (straight lines) BA, AC.** Thus, on 
«-cptcxoA</n| being dropped, the expression would become first ^ M rtDv BAT Tw^fo, then 
4 ^6 BAT ywla, and finally ^ inr6 BAP, without ytatda, as we rqrolarly find it in Euclid. 

17. if the triangle be applied to..., 33. coincide. The difference between the 
technical use of the passive i^apfU^taBcu "to be a^piitd {to)" and of the active i^apfUftuf 
"to eeindde (with) '^ has been noticed above (note on Common Notion 4, pp. 334 — 5). 

J 3. [For \i, when B coincides... ^6. coincide with EF]. Heiberg {P^troHpcminmau 
fid in Hermis, xxxviii., 1903, p. 50) has pointed out, as a conclusive reason for regarding 
these words as an early interpolation, that the text of an-Nairld {Codex Leidensis ^99, 1, ed. 
Besthom-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.B.D., 
which shows that they constitute a scholium only. They were doubtless added bjjr some 
commentator who thought it necessary to explain the immediate inference that, smce B 
coincides with E and C with F, the straight line BC coincides with the straight line RF, 
an inference which really follows from the definition of a straight line and Post. 1 ; and no 
doubt the Postulate that " Two straight lines cannot enclose a space " (afterwards placed 
among the Common Notions) was interpolated at the same time. 

44. Therefore etc. Where (as here) Euclid's conclusion merely repeats the enunciation 
( word for word, I shall avoid the repetition and write " Therefore etc.*' simply. 

I In the note on Common Notion 4 I have already mentioned that Eudid 

^ obviously used the method of superposition with reluctance, and I have given, 

after Veronese for the most part, the reason for holding that that method is 

not admissible as a theoretical means of proving equality, although it may be 

of use as 9^ practical test, and may thus furnish an empirical basis on which to 

found a postulate. Mr Bertrand Russell observes {Principles of Maihemaiics 

I. p. 405) that Euclid would have done better to assume i. 4 as an axiom, as 

. is practically done by Hilbert {Grundlagen der Geometric^ p. 9). It may be 

( / that Euclid himself was as well aware of the objections to the method as are 

i ' his modem critics ; but at all events those objections were stated, with almost 

\^ equal clearness, as early as the middle of the i6th century. Peletarius 

(Jacques Peletier) has a long note on this proposition (In EucHdis EUmemta 

j geometrica demonstrationum libri sex^ i557)> in which he observes that, if 

I' superposition of lines and figures could be assumed as a method of proof^ the 

\ whole of geometry would be full of such proofs, that it could equally well have 

4 been used in i. 2, 3 (thus in i. 2 we could simply have supposed the line taken 

s up zxA placed at the point), and that in short it is obvious how far removed the 

' method is from the dignity of geometry. The. theorem, he adds, is obvious in 

itself and does not require proof ; although it is introduced as a theorem, it 

would seem that Euclid intended it rather as a definition than a theorem, '^for 

I cannot think that two angles are egual unless I have a conception of what 

equality of angles is.'' Why then did Euclid include the proposition among 

theorems, instead of placing it among the axioms ? Peletanus makes the best 

excuse he can, but concludes thus: '' Huius itaque propositionis veritatem non 

aliunde quam a communi iudido petemus; cogitabimusque figuras figuris 

superponere, Mechanicum quippiam esse: intelUgere verb, id demum esse 

Mathemadcum.'' 

Expressed in terms of the modem systems of Congruence-Axioms referred 
to in the note on Common Notion 4, what Euclid really assumes amounts to 
»' the following : 

(i) On the line DE^ there is a point E^ on either side of 27, such that AB 
is equal to DE. 



i 



250 BOOK I [1.4 

(2) On either side of the ray DE there is a ray DF such that the angle 
EDF'v^ equal to the angle BAC. 

It now follows that on DF there is a pcnnt F such that DF is equal 
to^C. 

And lastly (3), we require an axiom from which to infer that the two 
remaining angles of the triangles are respectively equal and that the bases aie 
equal. 

I have shown above (pp. 229—330) that Hilbert has an axiom stating the 
equality of the remaining angles simply, but proves the equality of the b^^ses. 

Another alternative is that of Pasdi ( Variesungm uber neuert Geomdrk^ 
p. 109) who has the following ''Grundsatz": 

If two figures AB and FGH are given (FGH not being contained in a 
straight length), and AB^ FG are congruent and if a plane surfoce be laid 
through A and B^ we can specify in this plane surface, produced if necessary, 
two points C, Z>, neither more nor less, such that the figures ABC and ABD 
are congruent with the figure FGH^ and the straight Tine CD has with the 
straight line AB or with AB produced one point common. 

I pass to two points of detail in Euclid's proof: 

(i) The inference that, since B coincides with E^ and C with F^ the 
bases of the triangles are wholly coincident rests, as expressly stated, on the 
impossibility of two straight lines enclosing a space, and therefore presents no 
difficulty. 

But (2) most editors seem to have fiuled to observe that at the very 
beginning of the proof a much more serious assumption is made without any 
explanation whatever, namely that, if ^^ be placed on D^ and AB on DE^ the 
pomt B will coincide with ^, because AB is equal to DE. That is, the 
converse of Common Notion 4 is assumed for straight lines. Produs merely 
observes, with regard to the converse of this Common Notion, that it is only 
true in the case of things "of the same form " (6/ioc(S^), which he explains as 
meaning stnught lines, arcs of one and the same circle, and angles '' contained 
by lines simiku: and similarly situated" (p. 241, 3 — 8^. 

Savile however saw the difficulty and grappled with it in his note on the 
Common Notion. After stating that all straight lines with two points common 
are congruent between them (for otherwise two straight lines would enclose a 
space), he argues thus. Let there be two straight lines AB^ DE^ and let A be 
placed on 2?, and AB on DE. Then B will coincide with E. For, if not, 
let B fall somewhere short of E or beyond E ; and in either case it will follow 
that the less is equal to the greater, which is impossible. 

Savile seems to assume (and so apparently does Lardner who gives the 
same proof) that, if the straight lines be "applied," B will fall somewhere on 
DE or DE produced. But the ground for this assumption should surely be 
stated ; and it seems to me that it is necessary to use, not Postulate i alone, 
nor Postulate 2 alone, but both, for this purpose (in other words to assume, 
not only that two straight lines cannot enclose a space^ but also that two straight 
lines cannot have a common segment). For the only safe course is to place A 
upon D and then turn AB about D until some point on AB intermeduite 
between A and B coincides with some point on DE. In this position AB an^ 
DE have two points common. Then Postulate i enables us to infer that the 
straight lines coincide between the two common points, and Postulate 2 that 
they coincide beyond the second common point towards B and E. Thus the 
straight lines coincide throughout so fieur as both extend; and Savile's argument 
then proves that B coincides with E. 



J 




i.,5] PROPOSITIONS 4, 5 2^1 

Proposition 5. 

In isosceles triangles the angles at the base are equal to one 
another^ and, if the equal straight lines be produced further, 
the angles under the base will be equal to one another. 

Let ABC be an isosceles triangle having the side AB 
5 equal to the side AC\ 

and let the straight lines BD, CE be produced further in a 
straight line with AB, AC. [Post 2] 

1 say that the angle ABC is equal to the angle ACB^ and 
the angle CBD to the angle BCE. 
10 Let a point F be taken at random 
on BD] 

from AE the greater let AG he cut off 
equal to AF the less ; [i. 3] 

and let the straight lines FC, GB be joined. 

[Post i] 

15 Then, since AF is equal to -^G^ and 
^^to^C, 

the two sides FA^ AC are equal to the 
two sides GA, A By respectively ; 
and they contain a common angle, the angle FAG. 
20 Therefore the base FC is equal to the base GB, 

and the triangle AFC is equal to the triangle AGB, 
and the remaining angles will be equal to the remaining angles 
respectively, namely those which the equal sides subtend, 

that is, the angle ACF to the angle ABG, 
25 and the angle AFC to the angle AGB. [i. 4] 

And, since the whole AFis equal to the whole AG, 
and in these AB is equal to AC, 
the remainder BF is equal to the remainder CG. 
But FC was also proved equal to GB ; 
30 therefore the two sides BF, FC are equal to the two sides 
CG, GB respectively ; 
and the angle BFC is equal to the angle CGB, 

while the base BC is common to them ; 
therefore the triangle BFC is also equal to the triangle CGB, 
35 and the remaining angles will be equal to the remaining 



[ 



353 BOOK I [l 5 

angles respectively, namely those which the equal sides 
subtend ; 

therefore the angle FBC is equal to the angle GCB^ 
and the angle BCF to the angle CBG. 
¥> Accordingly, since the whole angle ABG was proved 
equal to the angle ACF, 

and in these the angle CBG is equal to the angle BCF, 

the remaining angle ABC is equal to the remaining angle 
ACB; 

15 and they are at the base of the triangle ABC. 

But the angle FBC was also proved equal to the angle GCB ; 

and they are under the base. 

Therefore etc. Q. E. d. 

1. the equal straight lines (meaniiig the equal sides). Cf. note on the amilar 
expression in Frop. 4, lines 1, $. 

la Let a point P be taken at random on BD, c2X4^^ M rft BA rvx^ tfit^wSbr H E, 
where rvx^v aii/uw means "a chance point.** 

17. the two sides FA, AC are equal to the two sides QA, AB respccthreljr, Mo 
ol Zl, AT ival nut HA, AB Krai tlaiw ixmripa inrip^. Here, uid in nnmberlest later 
passages, I have inserted the word "ades** for the reason given in the note on I. i, fine so. 
It would have been permissible to sopply either '* straight lines" or "ades**; but on the 
whole "sides '* seems to be more in accordance with the phrasedogjr of I. 4. 

33. the base BC is common to them, i.e., apparently, common to the atngUSf as 
the odrc^v in fidatt ai^rwr icoirii can only refer to ytpUt and yvAa preceding. Simson wrote 
**and the base ^C is common to the two triangles BFC^ CGB*'\ Todhunter left out these 
words as being of no use and tending to perplex a beginner. But Euclid evidently chose 
to quote the conclusion of I. 4 exactly ; the first phrase of that conclusion is that the bases 
(of the two triangles) are equal, and, as the equal bases are here the sams base, Euclid 
naturally substitutes the word "common** for "equal." 

48. As ** (deing) what it was required to prove '* (or " do '*) is somewhat long, I shall 
henceforth write the time-honoured "Q. E. D.' and "Q. E. F." for Swtp idu 9€t^ and 9rtp 

According to Proclus (p. 250, 20) the discoverer of the fact that in any 
isosceles triangle the angles at the base are equal was Thales, who however 
is said to have spoken of the angles as being similar^ and not as being equal, 
(Cf. Arist De caelo iv. 4, 31 1 b 34 irpos hyjom -^^wm ^v€rai ^cpoficyoy where 
egua/ angles are meant.) 

A pre-Euclidean proof of I. 5. 

One of the most interesting of the passages in Aristotle indicating differences 
between Euclid's proofs and those with which Aristotle was fieunilmr, in other 
words, those of the text-books immediately preceding Euclid's, has reference to 
the theorem of i. 5. The pissage (Anal. Prior, i. 24, 41 b 13—22) is so 
important that I must quote it in full. Aristotle is illustrating the fact that in 
any syllogism one of the propositions must be affirmative and universal 
(ffoMXov). ''This," be says, ''is better shown in the case of geometrical 
propositions ^ i^ roTs &aypafi/Aao-tv), e.g. the proposition that the angles at the 
base of an isosceles triangle are equal, 

"For let ^, ^ be drawn [i.e. joined] to the centre. 




I. S] PROPOSITION 5 «53 

''If, then, we assumed (i) that the angle AC [Le. ^ + C] is equal to die 
angle BD [Le. B-^D^ without asserting generally 
that the angles of semidrcUs are equals and again 
(2) that the angle C is equal to the angle D without 
making the further assumption that the two angles of 
ali segments are equals and if we then inferred, lastly, 
that, since the whole angles are equal, and equal 
angles are subtracted from them, the angles which 
remain, namely ^, ^ are equal, we should commit 
a petitio prinapii'^ 

Some peculiarities of phraseology will be observed 
in this passage. 

(i) A^ B are said to be drawn (4yfiiyai) to the centre (of the cirde of 
which the two equal sides are radii) asif A^B were not the angular points but 
the sides or the radii themselves. There is at least one pandlel for this in 
Euclid (cf. IV. 4). 

(2) "The angle AC*' is the angle which is the sum of A and C, and A 
means here the angle at A of the isosceles triangle shown in the figure, and 
afterwards spoken of by Aristotle as £, while C is the " mixed " angle between 
AB and the circumference of the smaller segment cut off by it 

(3) The ''angle of a semicircle" (i.e. the "angle" between the diameter 
a^d the circumference, at the extremity of the diameter) and the "angle ^ a 
segment" appear in Euclid in. 16 and in. Def. 7 respectively, obviously as 
survivals from earlier text-books. ■ .^.-^ 

But the most significant facts to be gathered from the extract are that in 
the text-books whidb preceded Euclid's " mixed " angles played a much more 

important part than they do with Euclid, and, in particular, that at least two "^ 

propositions concerning such angles appeared quite at the beginning, namdy "x 

the propositions that the (mixed) angles of semicircles are equal and that the two .% 

(fkixed) angles of any segment of a circle are equal. The wording of the first . ;: ; 

of the two propositions is vague, but it does not necessarily mean more dian ' V-^ 

that the two (mixed) angles in one semicircle are equal, and I know of no -J; 

evidence going to show that it asserts that the angle of any one semicircle is ^^ 
equal to the angle of any other semicircle (of different size). It is quoted in _ "^ 

the same form, " because the angles of semicircles are equal," in die Latin n^ 

translation from the Arabic of Heron's Catoptrica^ Prop. 9 (Heron, VoL VL^ '^ 

Teubner, p. 334), but it is only inferred that the different radii of one drde *^ V? 

make equal "angles" with the circumference ; and in the similar proposition --^) 

of the Pseudo-Euclidean Catcptrica (Euclid, Vol. vii., p. 294) ai4;les of the > 

same sort in one circle are said to be equal "because they are (angles) of "'^ 

a semicircle." Therefore the first of the two propositions may be cmly a ' \ 

particular case of the second. . t> 

But it is remarkable enough that the second proposition (that the two "^ 

" CMgles of any segment of a circle are equal) should, in earlier text-books, have ;^ : 

been placed before the theorem of Eucl. i. 5. We can hardly suppose it to ^> 
have been proved otherwise than by the superposition of the semicndes into 

which the circle is divided by the diameter which bisects at right angles the _^ A 

base of the segment; and no doubt the proof would be closely connected 1^ "* 
that of Thales' other proposition that any diameter of a circle bisects it, which 
must also (as Proclus indicates) have been proved by superposing one of the 
two parts upon the other. 

It is a natural inference fix>m the passage of Aristode that Euclid's proof of 




aS4 BOOK I D- 5 

I. 5 was his own, and it would thus appear that his innovations as regards 
order of propositions and methods of proof began at the very threshold of the 
subject 

Proof without producing the sides. 
In this proof, given by Proclus (pp. 248, a a — 349, 19), D and E are taken 
on AB^ AC, instead of on AB, Atprodttad, so that AD, AEwn, equal The 
method of proof is of course exactly like Euclid's, but it does not establish the 
equality of the angles beyond the base as well. 

Pappus' proof. 

Proclus (pp. 249, 20—250, 12) says that Pappus proved the theorem in a 
still shorter manner without the help of any construction whatever. 

This very interesting proof is given as follows : 

" Let ABC be an isosceles triangle, and AB equal to 
AC. 

Let us conceive this one triangle as two triangles, and let 
us arfl;ue in this way. 

Smce AB is equal to AC, and AC to AB, 
the two sides AB, AC are equal to the two sides AC, AB. 

And the angle BAC is equal to the angle CAB, for it is 
the same. 

Therefore all the corresponding parts (in the triangles) are equal, namely 

BC to BC, 

the triangle ABC to the triangle ABC (Le. ACB), 

the angle ABC to the angle ACB, 

and the angle ACB to the angle ABC, 

(for these are the angles subtended by the equal sides AB, AC. 

Therefore in isosceles triangles the angles at the base are equal." 

This will no doubt be recognised as the foundation of the alternative 
proof frequently given by modem editors, though they do not refer to Pappus. 
But they state the proof in a different form, the common method being to 
suppose the triangle to be taken up, turned over, and placed again upon stsetf, 
after which the same considerations of congruence as those used by Euclid in 
I. 4 are used over again. There is the obvious difficulty that it supposes the 
triangle to be taken up and at the same time to remain where it is. (Cf. 
Dodgson's humorous remark upon this, Euclid and his modem Rivals, p. 47.) 
Whatever we may say in justification of the proceeding (e.g. that the triangle 
may be supposed to leave a trace), it is really equi^ent to assuming &e 
construction (hypothetical, if you will) of another triangle equal in all respects 
to the given triangle ; and such an assumption is not in accordance with 
Euclid's principles and practice. 

It seems to me that the form given to the proof by Pappus himself is by &r 
the best, for the reasons (i) that it assumes no construction of a second 
triangle, real or hypothetiad, (3) that it avoids the distinct awkwardness 
involved by a proof which, instead of merely quoting and applying the result 
of a previous proposition, repeats, with reference to a new set of data, the 
process by which that result was established. If it is asked how we are to 
realise Pappus' idea of two triangles, surely we may answer that we keep to one 
triangle and merely view it in two aspects. If it were a question of helping a 
b^;inner to understand this, we might say that one triangle is the triangle 



\ 

t 

J 



I 



I. s, 6] PROPOSITIONS 5, 6 255 

looked at in front and that the other triangle is the same triangle looked at 
from behind] but even this is not really necessary. 

Pappus' proof, of course, does not include the proof of the second part of 
the proposition about the angles under the base, and we should still have to 
establish this much in the same way as Euclid does. 

Purpose of the second part of the theorem. 

An interesting question arises as to the reason for Euclid's insertion of the 
second part, to which, it will be observed, the converse proposition i. 6 has 
nothing corresponding. As a matter of fact, it is not necessary for any 
subsequent demonstration that is to be found in the original text of Euclid, 
but only for the interpolated second case of i. 7; and it was perhaps not 
unnatural that the undoubted genuineness of the second part of i. 5 convinced 
many editors that the second case of i. 7 must necessarily be Euclid's also. 
Proclus' explanation, which must apparently be the right one, is that the 
second part of i. 5 was inserted for the purpose of fore-arming the learner 
against a possible objection (Ivoroo-ic), as it was technically called, which might 
be raised to i. 7 as given in the text, with one case only. The objection would, 
as we have seen, take the specific ground that, as demonstrated, the theorem 
was not conclusive, since it did not cover all possible cases. From this point 
of view, the second part of l 5 is useful not only for i. 7 but, according to 
Proclus, for i. 9 also. Simson does not seem to have grasped Proclus' 
meaning, for he says : " And Proclus acknowledges, that the second part of 
Prop. 5 was added upon account of Prop. 7 but gives a ridiculous reason for 
it, 'that it might afford an answer to objections made against the 7th,' as if the 
case of the 7th which is left out were, as he expressly makes it, an objection 
against the proposition itself." 



Proposition 6. 

If in a triangle two angles be equal to one another^ the 
sides which subtend the equal angles will also be equal to one 
another. 

Let ABC be a triangle having the angle ABC equal to 
the angle ACB\ 

I say that the side AB is also equal to the 
^ side AC. 

j For, if AB is unequal to AC, one of them is 

greater. 

Let AB be greater; and from AB the 
greater let DB be cut off equal to -^C the less ; 

let DC be joined. 

Then, since DB is equal to AC, 
and BC is common, 

the two sides DB, BC are equal to the two sides AC, 
CB respectively ; 




956 BOOK I [l6 

and the angle DBC is equal to the angle ACB ; 

therefore the base DC is equal to the base AB^ 
and the triangle DBC will be equal to the triangle ACB^ 

the less to the greater : 
which is absurd 

Therefore AB is not unequal to AC; 
it is therefore equal to it 
Therefore etc. 

Q. E. D. 

Euclid assumes that, because Z> is between A and B, the triangle DBC 
13 less than the triangle ABC. Some postulate is necessary to justify diis 
tadt assumption; considering an angle less than two right angles, say the 
angle ACB in the figure of tiie pr(qx)sition, as a cluster df rays issuing fiom 
C and bounded by the rays CA^ CB^ and joining AB (where A,B$xe any 
two points on CA^ CB respectively), we see that to each successive ray taken 
in the direction from CA to CB there corresponds one point on AB in whidi 
the said ray intersects AB^ and that all the points on AB taken in <mler from 
A to B correspond univocally to all the ra^ taken in order frcun CA to 
CB, each point namely to the ray intersectmg AB in the point 

We have here used, for the &rst time in the ElementSf the method of 
reductio ad absurdum^ as to which I would refer to the section above (pp. 136, 
140) dealing with this among other technical terms. 

This proposition also, being the converse of the preceding proposition, 
brings us to the subject of 

Geometrical Conversion. 

This must of course be distinguished from the logical conversion of a 
proposition. Thus, from the proposition that all isosceles triangles have the 
angles opposite to the equal sides ecjual, logical conversion would only enable 
us to conclude that some triangles with two angles equal are isosceles. Thus 
I. 6 is the geometrical, but not the logical, converse of i. 5. On the other 
hand, as De Morgan points out {Companion to the Almanac, 1849, p. 7), L 6 is 
a purely ^Ss^o/ deduction from i. 5 and i. 18 taken together, as is i. 19 aba 
For the general argument see the note on i. 19. For die present proposition 
it is enough to state the matter thus. Let X denote the class of triangles 
which have the two sides other than the base equal, Y the class of triangles 
which have the base angles equal \ then we may call non-A' the class of 
triangles having the sides other than the base unequal, non- Y the class of 
triangles having the base angles unequal 

Thus we have 

All AT is K, [i. s] 

All lion-X is non-K; [i. 18] 
and it is a purely logical deduction that 

All y is AT. [i. 6] 

According to Produs (p. 352, 5 sqq.) two forms of geometrical conversion 
were distinguished. 

(i) The leading form {jrpmfyoviUrq), the oonvemoTi par excellence (1I1 KVfimi 



1 



f 



1.6] PROPOSITION 6 15? 

ayruFTfio^, is the complete or simple conversion in which the hypothesis 
and the conclusion of a theorem change places exactly, the conclusion of the 
theorem bein^ the hypothesis of the converse theorem, which again establishes, 
as its conclusion, the hypothesis of the original theorem. The relation between 
the first part of i. 5 and i. 6 is of this character. In the former the hypothesis 
is that two sides of a triangle are equal and the conclusion is that the angles 
at the base are equal, while the converse (i. 6) starts from the hypothesis that 
two angles are equal and proves that the sides subtending them are equal. 

(2) The other form of conversion, which we may call partial^ is seen 
in cases where a theorem starts from two or more hypotheses combined into 
one enunciation and leads to a certain conclusion, liter which the converse 
theorem takes this conclusion in substitution for one of the hypotheses of 
the original theorem and from the said conclusion along with the rest of the 
original hypotheses obtains, as its conclusion, the omitted hypothesis of the 
original theorem, i. 8 is in this sense a converse proposition to i. 4 ; for i. 4 
takes as hypotheses (i) that two sides in two triangles are respectively equal, 
(3) that the included angles are equal, and proves (3) that the bases are equal, 
while I. 8 takes (i) and (3) as hypotheses and proves (2) as its conclusion. It 
is clear that a conversion of the leading type must be unique, while there 
may be many partial conversions of a theorem according to the number of 
hjrpotheses from which it starts. 

Further, of convertible theorems, those which took as their hypothesis 
the gtnus and proved a property were distinguished as the leading theorems 
(irpoi;yov/tcva), while those which started from the property as hypothesis 
and described, as the conclusion, the genus possessing that property were the 
converse theorems, i. 5 is thus the leading theorem and i. 6 its converse, 
since the genus is in this case taken to be the isosceles triangle. 

Converse of second part of I. 5. 

Why, asks Proclus, did not Euclid convert the second part of i. 5 as well ? 
He suggests, properly enough, two reasons: (i) that the second part of i. 5 
itself is not wanted for any proof occurring in the original text, but is only put 
in to enable objections to the existing form of later propositions to be met, 
whereas the converse is not even wanted for this purpose \ (2) that the converse 
could be deduced from i. 6, if wanted, at any time after we have passed i. 13, 
which can be used to prove that, if the angles formed by producing two sides 
of a triangle beyond the base are equal, the base angles themselves are equal. 

Proclus adds a proof of the converse of the second part of i. 5, Le. of the 
proposition that, if the angles formed by producing two 
sides of a triangle beyond the base are equal, the triangle 
is isosceles; but it runs to some length and then only 
effects a reduction to the theorem of i. 6 as we have it. 
As the result of this should hardly be assumed, a better 
proof would be an independent one adapting Euclid's 
own method in i. 6. Thus, with the construction of i. 5, 
we first prove by means of i. 4 that the triangles BFC^ 
CGB are equal in aU respects, and therefore that FC is 
equal to GB^ and the angle BFC equal to the angle CGB. 
Then we have to prove that AF^ AG sure equal. If they 
are not, let AF be the greater, and from FA cut off FH equal to GA. 
Join CH. 

H. E. 17 




JS8 hOOlt t ti.6,7 

Then we have, in the two triaz^les HFC^ AGB^ 

two sides HF^ FC equal to two sides AG^ GB 
and the angle ^/C equal to the angle AGB. 

Therefore (i. 4) the triangles HFC^ AGB are equal But the triangles 
BFC, CGB are also equal 

Therefore (if we take away these equals respectively) the triangles HBQ 
ACB are equal: which is impossible. 

Therefore AF^ AG are not unequal. 

Hence AF\& equal to AG and, if we subtract the equals BF^ CG respec- 
tively, AB is equal to AC. 

This proof is found in the commentary of an-Naiibl (ed. Besthom-Heibeigi 
p. 61 ; ed. Curtze, p. 50). 

Alternative proofs of I. 6. 

Todhunter points out that i. 6, not being wanted till 11. 4, could be 
postponed till later and proved by means of i. 26. Bisect the angle BAC 
by a straight line meeting the base at 2>. Then die triangles ABD^ ACD 
are equal in all respects. 

Another method depending on i« 26 is given by an-Nairld after that 
proposition. 

Measure equal lengths BD^ CE along the sides BA^ CA. 
Join BE, CD. 

Then [i. 4] the triangles DBC, ECB are equal in all 
respects; 

therefore £B, DC are equal, and the angles BEC, CDB 
are equal. 

The supplements of the latter angles are equal [1. 13], 
and hence the triangles ABE, ACD luive two angles equal respectively and 
the side BE equal to the side CD. 

Therefore [1. 26] AB is equal to AC. 

Proposition 7. 

Given two straight, lines constructed on a straight line 
{from its extremities) and fneeting in a point, there cannot be 
constructed on the same straight line {from its extremities\ 
and on the same side of it, two other straight lines meeting tn 

5 another point and equal to the former two respectively, namely 
each to that which has the same extremity with it. 

For, if possible, given t^o straight lines AC, CB con- 
structed on the straight line AB and meeting 
at the point C, let two other straight lines 

10 AD, DB be constructed on the same straight 
line AB, on the same side of it, meeting in 
another point D and equal to the former two 
respectively, namely each to that which has 
the same extremity with it, so that CA is 

15 equal to DA which has the same extremity A with it, and 





,1 



L 7] PROPOSITIONS 6, 7 ^59 



CB to DB which has the same extremity B with it ; and let 
CD be joined. 

Then, since -^C is equal to ADy 

the angle ACD is also equal to the angle ADC; [i. s] 

therefore the angle ADC is greater than the angle DCB ; 

therefore the angle CDB is much greater than the ancfle 
DCB. 
\ Again, since CB is equal to DB, 
" \ the angle CDB is also equal to the angle DCB. 

■ |; But it was also proved much greater than it : 

which is impossible. 
Therefore etc. q. e. d. 



i! 



1—6. In an English translation of the enunciation of this proposition it is absolutely 
"...... . wnicn . . - 



I The reason is partly that the Greek enunciation is itself very elliptioU, and partly that some 
words used in it conveyed more meaning than the corresponding words in English do. 
Particularly is this the case with 06 ffvera^orrai M "there shall not be constructed upon," 
since ^wi^rmff9ai is the regular word for constructing a triangle in particular. Thus a Ureek 

' would easily understand vwraBi/faoirrai 4vl as meaning the construction of two Unts forming' 
a triangle on a given straight line as base ; whereas to "construct two straight lines on a 
straight line*' is not in English sufficiently de6nite unless we explain that they are drawn 
from the ends of the straight line to meet at a point. I have had the less hesitation in putting 
in the words "from its extremities" because they are actually used by Euclid in the somewhat 
similar enunciation of I. 91. 

How impossible a literal translation into English is, if it is to convey the meaning of the 
enunciation mtelligibly, will be clear from the following attempt to render literally: **On the 
same straight line there shall not be constructed two other straight lines equal, each to each, 
to the same two straight lines, (terminatixig) at different points on the same side, having the 
same extremities as the original straight lines *' (M r^f 0^1% tMtlat M rsut adroit eiStUut 

\tidp^ tA oMl Wpara ix^wm rtus ^ d^X^t ci^^eiait). 

i The reason whv Euclid allowed himself to use, in this enunciation, language apparently 

I so obscure is no doubt that the phraseology was traditional and therefore, vague as it was, 

I had a conventional meaning whidi the contemporary geometer well understood. This is 

I proved, I think, by the occurrence in Aristotle {Metearvlogiea lii. 5, 376 a 2 sqq.) of the very 

'same, evidently technical, expressions. Aristotle is there alluding to the theorem given by 

j Entodus from Apollonius' Plane Loci to the effect that, if /T, ^ be two fixed points and M 

such a variable point that the ratio of MH to MK is a given ratio (not one of equality^ the 

locus of A/ is a circle. (For an account of this theorem see note on vi. 3 below.) Now 

Aristotle says " The lines drawn up from /^, AT in this ratio cannot be constructed to two 

different points of the semicircle ^ ** (ol oj^r drd rQnf HK dMiY^/««yai ypoLtiftaX ^ rodry r^ 

Xiytf oi ovvTaSiiVQwrai roO i^l', f A iifUKWcktov vpbt AWo Kal dXXo ffii/ittow). 

If a paraphrase is allowed instead of a translation adhering as closely as possible to the 
original, Simson's is the best that could be found, since the fiiict that the straight lines form 
triimgles on the same base is really conveyed in the Greek. Simson*s enunciation is. Upon 
the same Aau, and on the same side of it^ there cannot be two triangles that have their sides 
which are terminated in one extremity of the base equal to one another^ and likewise those 
vihieh art terminated at the other extremity, Th. Taylor (the translator of Produs) attacks 
'imson's alteration as "indiscreet" and as detractmg from the beauty and accuracy of 
Jodid's enunciation which are enlarged upon by Proclus in his commentary. Yet, when 
Taylor says ** Whatever difficulty learners may find in conceiving this proposition abstractedly 
b CMily removed by its exposition in the figure," he really gives his case away. The fact is 
l|that Taylor, always enthusiastic over his author,, was nettled by Simson's sighting remarks 
on Proclus' comments on the proposition. Simson had said, with reference to Proclus' 
explanation of the bearing of the second part of i. 5 on i. 7, that it was not "worth while 

17 — 2 



36o fiook t [1.7 

to relate his trifles mt full length," to whidi Taylor retorts ''But Mr Simaoo was no 
philosopher ; and therefore the greatest part of these Commentaries must be oonsiderad fay 
nim as trifles, from the want of a phuoaophic genius to comprehend their meaning, and 
a taste superior to that of a nc/rv mmthmtUidam^ to discover their beauty and d^guoe.** 

so. It would be natural to insert here the step '*lmt the angle ACD is greater than the 
angle ^C/>. [C. A^. 5.]" 

1 1 . much greater, literally " greater by much " (voXXf /mI^) • Simsoo and thoae who 
follow him transUte : ^^much mwn tkm is the angfle BDC grmitr than the anrie BCD^^ 
but the Greek for this would have tobe veXX^ (or a-oXv) ^kWkw ^#n...#Mf^. nXXf fJOJm^ 
however, though used by Apollonius, Is not, apparently, found in Endid or Archimedes. 

Just as in 1. 6 we need a Postulate to justify tbeoreticaQy die statement that 
CD falls within the angle ACB^ so that the triangle DBC is less than the 
triangle ABQ so here we need Postulates which shall satisfy us as to the 
relative positions of CA^ CB^ CD on the one hand and of />C, DA^ DB 
on the other, in order that we may be able to infer that the angle BDC is 
greater than the angle ADC^ and the angle ACD greater than the angle BCD. 

De Morgan {op. at. p. 7) observes that 1. 7 would be made easy to 
beginners if they were first familiarised, as a common notion, with **if two 
magnitudes be equal, any magnitude greater than the one is greater than any 
magnitude less than the other." I doubt however whether a beginner would 
follow this easily ; perhaps it would be more easily apprehended in the form 
''if any magnitude A is greater than a magnitude B^ the magnitude A is 
(a fortiori) greater than any magnitude less than B or than any magnitudie 
equal to B." 

It has been mentioned already (note on i. 5) that the second case of l 7 
given by Simson and in our text-books generally is not in the original text 
(the omission being in accordance with Euclid's general practice of giving 
only one case, and tiiat the most difficult, and leaving the others to be worked 
out by the reader for himself). The second case is given by Produs as the 
answer to a possible objection to Euclid's proposition, which should assert that 
the proposition is not proved to be universally true, since the proof given does 
not cover all possible cases. Here the objector is supposed to contend that 
what Euclid declares to be impossible may still be possible if one pair of lines 
lie wholly within the other pair of lines; and the second part of i. 5 enables 
the objection to be refuted. 

If possible, let AD^ DB be entirely within the triangle formed by AC^ 
CB with ABy and let AC be equal to AD and BC \ 

to BD. 

Join CD^ and produce AC^ AD to E and F. 

Then, since AC is equal to AD^ 

the triangle ACD is isosceles, 

and the angles ECD^ FDC under the base are equal. 
But the angle ECD is greater than the angle BCD \ 

therefore the angle FDC is also greater than the angle 

BCD. 

Therefore the angle BDC is greater by far than the angle BCD. 

Again, since DB is equal to CB^ 
the angles at the base of the triangle BDC are equal, [i. 5 ^ 

that is, the angle BDC is equal to the angle BCD. 

Therefore the same angle BDC is both greater than and equal to the ai^le 
BCD: which is impossible. 

The case in which D CeJIs on AC or BC does not require proof. 




i 



I. 7, 8] PROPOSITIONS 7, 8 261 

I have already referred (note on i. i) to the mistake made by those 
editors who regard i. ^ as being of no use except to prove i. 8. What i. 7 
proves is that if[ in addition to the base of a triangle, the length of the side 
terminating at each extremity of the base is given, only one triangle satisfying 
these conditions can be constructed on one and the same side of the given 
base. Hence not only does i. 7 enable us to prove i. 8, but it supplements 
1. I and I. 22 by showing that the constructions of those propositions ^ve one 
triangle only on one and the same side of the base. But for i. 7 this could 
not be proved except by anticipating iii. 10, of which therefore i. 7 is the 
equivalent for Book i. purposes. Dodgson (Euclid and his modem Rivals^ 
pp. 194 — 5) puts it in another way. " It [i. 7] shows that, of all plane figures 
that can be made by hingeing rods together, the thrte-siAeA ones (and these 
only) are rigid (which is another way of stating the fact that there cannot be 
iw0 such figures on the same base). This is imalogous to the fact, in relation 
to solids contained by plane surfaces hinged together, that any such solid is 
rigid, there being no maximum number of sides. And there is a close analogy 
between i. 7, 8 and iii. 23, 24. These analogies give to geometry much of its 
beauty, and I think that they ousht not to be lost sight of." It will therefore 
be apparent how ill-advised are those editors who eliminate i. 7 altogether and 
rely on Philo's proof for i. 8. 

• Proclus, it may be added, gives (pp. 268, 19 — 269, 10) another explanation 
of the retention of i. 7, notwithstanding that it was apparently only required 
for I. 8. It was said that astronomers used it to prove that three successive 
eclipses could not occur at equal intervals of time, i.e. that the third could not 
follow the second at the same interval as the second followed the first ; and it 
was ai|;ued that Euclid had an eye to this astronomical application of the 
proposition. But, as we have seen, there are other grounds for retaining the 
proposition which are quite sufficient of themselves. 

Proposition 8. 

1/ two triangles have the two sides equal to two sides 

respectively, and have also the base equal to t/ie base, they will 

aba have the angles equal which are contained by the equal 

straight lines. 

\ Let ABC, DEF be two triangles having the two sides 

\AB, AC equal to the two sides 

^DE, DF respectively, namely 

.AB to DE, and AC to DF\ and 

^let them have the base BC equal 

to the base EF\ 

1 say that the angle BAC is 
also equal to the angle EDF. 

For, if the triangle ABC be 
applied to the triangle DEF, and if the point B be placed on 
[the point E and the straight line BC on EF, 
the point C will also coincide with F, 
because BC is equal to EF. 





263 BOOK I [l8 

Then, BC coinciding with EF^ 
BA. AC will also coincide with ED, DF\ 
» for, if the base BC coincides with the base EF, and the sides 
BA, AC do not coincide with ED, DF but fall beside them 
as EG, GF, 

then, given two straight lines constructed on a straight 

line (from its extremities) and meeting in a point, there will 

sshave been constructed on the same straight line (from its 

extremities^, and on the same side of it, two other straight 

lines meeting in another point and equal to the former 

two respectively, namely each to that which has the same 

extremity with it 

30 But they cannot be so constructed. [i. 7] 

Therefore it is not possible that, if the base BC be applied 

to the base EF, the sides BA, AC should not coincide with 

ED, DF; 

they will therefore coincide, 
35 so that the angle BAC will also coincide with the angle 
EDF, and will be equal to it 

If therefore etc. q. e. d. 

19. B A, AC. The text has here " BA, CA, " 

91. fall beside them. The Greek has the future, wa^taXk^qvvi, wapoXKirrm means 
" to pass by without touchiug,** ** to miss" or *' to deviate." 

As pointed out above (p. 257) i. 8 is a /or/Za/ converse of i. 4. 

It is to be observed that in i. 8 Euclid is satisfied with proving the equality 
of the vertical angles and does not, as in i. 4, add that the triangles are equal, 
and the remaining angles are equal respectively. The reason is no doubt (as 
pointed out by Proclus and by Savile afler him) that, when once the vertical 
angles are proved equal, the rest follows from i. 4, and there is no object in 
proving again what has been proved already. 

Anstotle has an allusion to the theorem of this proposition in MeteorologUa 
III. 3, 373 a 5 — 16. He is speaking of the rainbow and observes that, if equal 
ra3rs be reflected from one and the same point to one and the same point, the 
points at which reflection takes place are on the circumference of a circle. 
"For let the broken lines ACB, AFB, ADB be all reflected from the point 
A to the point B (in such a way that) AC, AF, AD are all equal to one 
another, and the lines (terminating) at B, Le. CB, FB, DB, are likewise all 
equal ; and let AEB be joined. It follows that the triangles are equal; for 
they are upon the equal (base) AEB*^ 

Heibe^ {Mathematisches nt Aristoteles, p. 18) thinks that the form of the 
conclusion quoted is an indication that in the corresponding proposition to 
Eud. I. 8, as it lay before Aristotle, it was maintained that the triangles were 
equal, and not only the angles, and "we see here therefore, in a clear example, 
how the stones of the ancient fabric were recut for the rigid structure of his 



18] 



PROPOSITION 8 



363 



I 




I 



Elements.^ I do not, however, think that this inference from Aristotle's 
language as to the form of the pre-Euclidean proposition is safe. Thus if we, 
nowadays, were arguing from the data in the passage of Aristotle, we should 
doubtless infer dir^y that the triangles are equal in all respects, quoting i. 8 
alone. Besides, Aristotle's language is rather careless, as the next sentences 
of the same passage show. " Let perpendiculars," 
he says, *'be drawn to AEB from the angles, CE 
from C, FE from ^and DE from D. These, then, 
are equal j for they are all in equal triangles, and 
in one plane; for all of them are perpendicular 
to AEB^ and they meet at one point E. There- 
fore the (line) drawn (through C, F, 2P) will be a 
circle, and its centre (will be) E'* Aristotle should 
obviously have proved that the three perpendiculars will meet at one point E 
on AEB before he spoke of drawing the perpendiculars CE^ FE, DE, 
This of course follows from their being " in equal triangles " (by means of 
EucL I. 26); and then, from the fact that the perpendiculars meet at one 
point on AB, it can be inferred that all three are m one plane. 

Philo'8 proof of I. 8. 

This alternative proof avoids the use of i. 7, and it is elegant ; but it is 
inconvenient in one respect, since three cases have to be distinguished. 
Proclus gives the proof in the following order (pp. 266, 15 — 268, 14). 

I^ ABC, DEF be two triangles having the sides A By -<4C equal to the 
sides DEy 2?^ respectively, and the base ^C equal to the base EF, 

Let the triangle ABC be applied to the triangle DEF, so that B is placed 
on E and BC on EF, but so that A falls on the opposite side of EF from D, 
taking the position G. Then C will coincide with F, since BC is equal to 
EF. 

Now FG will either be in a straight line with DF, or make an angle with 
it, and in the latter case the angle will either be interior (icara to ivro^) to the 
figure or exterior (icara ro licw). 

I. Let FG he in a straight line with 
DF 

Then, since DE is equal to EG, and 
DFG is a straight line, 

DEG is an isosceles triangle, and the 
angle at Z^ is equal to the angle at G. 

[I- s]. 

II. Let DFy FG form an angle interior to the figure 
Let DG be joined. 
Then, since DE, EG are equal, 

the angle EDG is equal to the angle 
EGD. 

Again, since DF\& equal to FG, 
the angle FDG is equal to the angle 
FGD. 

Therefore, by addition, 
the whole angle EDF is equal to the 
whole angle EGF, 





364 BOOK I [1.8, 



n 




III. Let DF^ FG form an angle txterwr to the figure. 

Let DG be joined. 

The proof poceeds as in the last case, 
except that subtraction takes the place of 
addition, and 

the remaining angle EDF is equal to the 
remaining angle EOF. 

Therefore in all three cases the angle 
EDF is equal to the angle EGF^ that is, 
to the angle >ff^C. 

It will be observed that, in accordance widi the practice of the Greek 
geometers in not recognising as an "angle " aiw angle not less than two right 
angles, the re-entrant angle of the quadrilateral JDEuF\% ignored and the ang^ 
DFG is said to be outside the figure. 

Proposition 9. 

To bisect a given rectilineal angle. 

Let the angle BAC be the ^ven rectilineal angle. 

It is then required to bisect it. 

Let a point D be taken at random on AB ; 
let AE be cut off from AC equal to AD ; [i. 3] 
let DE be joined, and on DE let the equilateral 
triangle DEFh^ constructed ; 
let .^/^ be joined. 

I say that the angle BAC has been bisected by the 
straight line AF. 

r or, since AD is equal to AE, 
and AF is common, 

the two sides DA, AF are equal to the two sides ^ 
EA, AF respectively. 

And the base DF is equal to the base EF\ 

therefore the angle DAF is equal to the angle EAF. j 

[I. 8] I 

Therefore the given rectilineal angle BAC has been | 

bisected by the straight line AF. Q. e. f. ^ 

It win be observed from the translation of this proposition that Euclid 
does not say, in his description of the construction, that the equilateral triangle | 
should be constructed on the side of DE opposite to ^ ; he leaves this to be < 
inferred from his figure. There is no particular value in Proclus' explanation \ 
as to how we should proceed in case any one should assert that he could not . 
recognise the existence of any space below DE. He supposes, then, the ! 
equilateral triangle described on the side of DE towards A, and hence has to -. 
consider three cases according as the vertex of the equilateral triangle fidls 
on A, above A or below it The second and third cases do not difier 




i 



I. 9] PROPOSITIONS 8, 9 265 

substantially from Euclid's. In the first case, where ADE is the equilateral 
triangle constructed on DE^ take any point ^on AD^ and from AE cut off 
AG equal to AF, Join DG^ J?^ meeting in H\ and 
join AH, Then ^^ is the bisector required. 

Proclus ako answers the possible AjecHon that 
might be raised to Euclid's proof on the ground that 
it assumes that, if the equilateral triangle m described 
on the side of DE opposite to A^ its vertex -F will lie 
within the angle BAC. The objector is supposed to 
argue that this is not necessary, but that ^ might fall 
eiSier on one of the lines forming the angle or outside 
it altogether. The two cases are disposed of thus. 

Suppose ^to fall as shown in the two figures below respectively. 

Then, since FD is equal to FE^ 
the angle FDE is equal to the angle FED, 

Therefore the angle CED is greater than the angle FDE ; and, in the 
second figure, a fortiori^ the angle CED is greater than the angle BDE. 

But, since ADE is an isosceles triangle, and the equal sides are produced. 





the angles under the base are equal, 

i.e., the angle CED is equal to the angle BDE, 

But the an^e CED was proved greater : which is impossible. 

Here then is the second case in which, in Proclus' view, the second part 
of I. 5 is useful for refuting objections. 

On this proposition Proclus takes occasion (p. 371, 15—19) to emphasize 
I the isjct that the given angle must be rectilineal^ since the bisection of any sort 

of angle (including angles made by curves with one another or with straight 
\ lines) is not matter for an elementary treatise, besides which it is questionable 

^whether such bisection is always possible. "Thus it is difficult to say 

iwhether it is possible to bisect the so-called horn-like angle " (formed by the 

prcumference of a circle and a tangent to it). 

Trisection of an angle. 

Further it is here that Proclus gives us his valuable historical note about 
ithe trisection of any acute angle, which (as well as the division of an angle in 
iny ^ven ratio) requires resort to other curves than circles, i.e. curves of the 
speaes which, after Geminus, he calls "mixed." "This," he says (p. 372, 
I — 12), "is shown by those who have set themselves the task of trisecting such 
a given rectilineal angle. For Nicomedes trisected any rectilineal anple by 
means of the conchoidal lines, the origin, order, and properties of iduch he 
has handed down to us, being himself the discoverer of their peculiarity. 
Others have done the same thing by means of the quadratrices of Hippias 
..and Nicomedes, thereby again using 'mixed' curves. ^ But others, starting 
I' from the Archimedean spirals, cut a given rectilineal angle in a given ratio." 



* ■ r 



366 



-JLJLl "•"■■> 



BOOK I 



[1.9 



(a) Trisection by means of the amckaid. 

I have already spoken of the ionchmd of Nioomedes (nole on Det a, 
pp. 160 — i) ; it remains to show how it could be nsed for trisecting an 
angle. Pappus explains this (iv. pp. 374 — 5) as follows. 

Let ABC be the given acute angle, and from any point A in AB draw 
A C perpendicular to BC. 




e O 

Complete the parallelogram FBCA and produce FA \xi z, point E luch 
that, if BE be joined, BE interapis between AC and AE a Ungtk DE fual 
to twice AB. 

I say that the angle BBC is one-third of the angle ABC 

For, joining A to G, the middle point of DE^ we have the three straUht 
. lines AG, DG, EG equal, and the angle AGD is double of the angle AED 
or EBC 

But DE is double of AB ; 
therefore AG, which is equal to DG, is equal to AB. 

Hence the angle AGD is equal to the angle ABG. 

Therefore the angle ABD is also double of the angle EBC; 
so that the angle EBC is one-third of the angle ABC, 

So bi Pappus, who reduces the construction to the drawing of BE so 
that DE shall be equal to twice AB. 

This is what the conchoid constructed with Bz&pole, AC2& directrix^ and 
distance equal to twice AB enables us to do ; for that conchoid cuts AE in 
the required point E. 

{b) Use of the quadratrix. 

The plural quadratrices in the above passage is a Hellenism for the 
singular quadratrix, which was a curve discovered by Hippias of Elis about 
4^ B.C. According to Produs (p. 356, 11) Hippias proved its properties ; 
and we are told (i) in the passage quoted above that Nicomedes also 
investi^ted it and Uiat it was used for trisecdi^ an angle, and (3) by Pappus 
(iv. pp. 250, 33 — 252, 4) that it was used by Dmostratus and Nicomedes and 
some more recent writers for squaring the circle, whence its name. It is 
described thus (Pappus iv. p. 252). 

Suppose that ABCD is a square and BED a quadrant of a circle with 
centred. 

Suppose (i) that a radius of the circle moves 
uniformly about A from the position AB to the 
position AD, and (2) that in the same time the 
line BC moves uniformly, always parallel to itself, 
and with its jextremity B moviiig along BA, from 
the position BC to the position AD. 

Then the radius AE and the moving line BC 
determine at any instant by their intersection a 
point F. 

The locus of ^is the quadratrix. 









p 


l 




'^ 




V ♦ 


1 1 


!iQ i 


> 



, I. 9, lo] PROPOSITIONS 9, 10 267 

1 The property of the curve is that, if ^ is any point, the arc BED is 
• to the arc ED as AB is to FH. 

i In other words, if ^ is the angle FAD^ p the radius vector AFand a the 
aide of the square, 

(p8in^)/a = ^/Jir. 

^ Now the angle EAD can not only be trisected but divided in any git*en 

I ratio by means of the quadratrix (Pappus iv. p. 286). 

I Fof let FH be divided at K in the given ratio. 

I Draw KL parallel to AD^ meeting the curve in L ; join AL and produce 

I it to meet the circle in N. 

Then the angles EAN^ NAD are in the ratio of FK to KH^ as is easily 
proved. 

(c) Use of the spiral of Archimedes. 

The trisection of an angle, or the division of an angle in any ratio, by 
means of the spiral of Archimedes is of course an equally simple matter. 
Suppose any angle included between the two radii vectores OA and OB of the 
^^Md, and let it be required to cut the angle AOB in a given ratio. Since 
the radius vector increases proportionally with the angle described by the 
sector which generates the curve (reckoned from the original position of the 
(rector coinciding with the initial line to the particular position assiuned), we 
iiave only to take the radius vector OB (the greater of the two OA^ 0B\ 
nark off OC along it equal to OA^ cut CB in the given ratio (at D say), and 
fhen draw the circle with centre O and radius OD cutting the spiral in E, 
uhen OE will divide the angle AOB in the required manner. 

I Proposition id. 

To bisect a given finite straight line. 

Let AB be the given finite straight line. 
Thus It is required to bisect the finite straight line AB. 
Let the equilateral triangle ABC be 
.K>nstructed on it, [i. i] 

ind let the angle ACB be bisected by the 
traight line CD ; [1. 9] 

\ I say that the straight line AB has 
■•en bisected at the point D. 
- For, since ACv& equal to CB, 
id CD is common, 

I the two sides AC, CD are equal to the two sides Bd 
'Z? respectively ; 

nd the angle A CD is equal to the angle BCD ; 

\ therefore the base AD is equal to the base BD. [i. 4} 

iTherefore the given finite straight line AB has been 
lisected at D. q. e. f. 







268 



BOOK I 



[I. lol 




[1.8; 



Apollonius, we are told (Produs, pp. 279, 16— 280, 4), bisected a stimight 
line AB by a construction like that of i. i. 
With centres A^ B, and raiUi AB^ BA respec- 
tively, two circles are described, intersecting in 
C, D. ]o\nmg CD, AC, CB, AD, DB, Apol- 
lonius proves in two steps that CD bisects AB. 

(i) Since, in the triangles A CD, BCD, 
two sides AC, CD are equal to two sides 
BC, CD, 
and the bases AD, BD are equal, 
the angle A CD is equal to the angle 
BCD. 

(2) The latter angles being equal, and AC being equal to CB^ while C£ 
is common, 

the equality of AE, EB follows by l 4. 

The objection to this proof is that, instead of assuming the bisection o 
the angle ACB, as already effected by i. ^ ApoUonius goes a step fiuthe» 
back and embodies a construction for bisecting the ang^ That iS| bt 
unnecessarily does over again what has been done before, which is open to 
objection from a theoretiad point of view. 

Proclus (pp. 277, 25 — 279, 4) warns us against being moved by thisi 
proposition to conclude that geometers assumed, as a preliminary hypothesis. | 
that a line is not made up of indivisible parts (If Vip«r). This might bt| 
argued thus. If a line is made up of indivisibles, there must be in a finitti 
line either an odd or an even number of them. If the number were odd,! 
it would be necessary in order to bisect the line to bisect an indivisible (the) 
odd one). In that case therefore it would not be possible to bisect a straight 
line, if it is a magnitude made up of indivisibles. But, if it is not so made 
up, the straight line can be divided ad infinitum or without limit (hr cnrcipor 
Staipciroi). Hence it was argued (^oo-u'), says Proclus, that the divisibility 
of magnitudes without limit was admitted and assumed as a geometrical 
principle. To this he replies, following (jeminus, that geometers did indeeci 
assume, by way of a common notion, that a continuous magnitude, Le. 
magnitude consisting of parts connected together (ounififMHtfr), is diviribli 
(8uup«Tov). But infinite divisibility was not assumed by them ; it was pravei 
by means of the first principles applicable to the case. **For when," h 
sa3rs, ''they prove that the incommensurable exists among magnitudes, an 
that it is not all things that are commensurable with one another, wlu 
else will any one say that they prove but that every ma^tude can \ 
divided for ever, and that we sludl never arrive at the mdivisible, tb* 
is, the least common measiure of the magnitudes? This then is matto* . 
demonstration, whereas it is an axiom that everything continuous is divisible 
so that a finite continuous line is divisible. The writer of the Elemeni 
bisects a finite straight line, starting from the latter notion, and not fix»m an 
assumption that it is divisible without limit ** Proclus adds that the propositio 
may also serve to refute Xenocrates' theory of indivisible lines (aro^ ypofifjuai 
The aigument given by Proclus to disprove the existence of indivisible linr 
is substantially that used by Aristotle as r^;ards magnitudes generally (c 
PAysics VI. I, 231 a 21 sqq. and especially vi. 2, 233 b 15 — 32). 




Ill] PROPOSITIONS lo, II 269 

Proposition ii. 

I To draw a straight line at right angles to a given straight 

line front a given point on it. 

Let AB be the given straight line, and C the given point 
on it. 

Thus it is required to draw from the point C a straight 
. line at right angles to the straight 
\ line AB. 

I Let a point D be taken at ran- 
\ dom on AC\ 

^ let CE be made equal to CD ; [i. 3] 
► on DE let the equilateral triangle 
\FDE be constructed, • [i. i] 
I and let FC be joined ; 

I say that the straight line FC has been drawn at right 
I angles to the given straight line AB from C the given point 
Jon it. 

• For, since DC is equal to CE, 
rCF is common, 
the two sides DC, CF are equal to the two sides EC, 
I ' CF respectively ; 

I and the base DF is equal to the base FE ; 

therefore the angle DCF is equal to the angle ECF\ 

['• «] 
and they are adjacent angles. 

But, when a straight line set up on a straight line makes 

the adjacent angles equal to one another, each of the equal 

ingles is right ; [Def. 10] 

therefore each of the angles DCF^ FCE is right 

Therefore the straight line CF has been drawn at right 

ngles to the given straight line AB from the given point 

r on it. 

Q. E. F. 

10. let CE be made equal to CD. The verb is ntlrSw which, as well as the other 
barts of ccifuu, is constantly used for the passive of rWriiu " to fiiace^ ; and the latter word 
is constantly used in the sense of making, e.g., one straight line equal to another straight line. 

De Morgan remarks that this proposition, which is " to bisect the angle 
inade by a straight line and its continuation " [Le. a flat angle], should be a 
particular case of i. 9, the constructions being the same. Tms is certainly 



270 



BOOK I 



[i. lit IS 



worth noting, though I doubt the advantage of rearranging the propositions 
in consequence. 

Apollonius gave a construction for this proposition (see Proclus, p. 282, 8) 
differing from Euclid's in much the same way as his construction for bisecting 
a straight line differed from that of i. 10. Instead of assuming an equilatenl 
triangle drawn without repeating the process of i. i, Apollonius takes D and 
E equidistant from C as in Euclid, and then draws circles in the manner of 




I. I meeting at F. This necessitates proving again that DFv^ equal to FE\ 
whereas Euclid's assumption of the construction of i. i in the wc^ *' let the 
equilateral triangle FDE be constructed " enables him to dispense with the 
drawing of circles and with the proof that DF is equal to FE at the same 
time. While however the substitution of Apollonius* constructions for l 10 
and II would show faulty arrangement in a theoretical treatise like Eudid's, 
they are entirely suitable for what we call practical geometry, and such may 
have been Apollonius' object in these constructions and in his alternative for 
I. 23. 

Proclus f^ves a construction for drawing a straight line at right angles to 
another straight line but from one end of it, instead of from an intermediate 
point on it, it being supposed (for the sake of argument) that we are not 
permitted to produce the straight line In the commentary of an-NaiiM (ed. 
Besthom-Heiberg, pp. 73 — 4; ed. Curtze, pp. 54 — 5) this constructiv 
attributed to Heron. 

Let it be required to draw from A a straight line at right angles to AB. . 

On AB take any point C, and in the manner of the proposition draw CE^ 
at right angles to AB, 

From CE cut off CD equal to AC^ bisect the 
angle ACE by the straight line CF^ [i. 9] 

and draw DF at right angles to CE meeting CF 
in F Join FA. 

Then the angle FAC will be a right angle. 

For, since, in the triangles ACF^ DCF^ the 
two sides AC^ CF are equal to the two sides 
DC^ CF respectively, and the included angles 
ACF^ DCFzx^ equal, 

the triangles are equal in all respects. [i. 4 

Therefore the angle at A is equal to the angle at Z>, and is accordingly ' 
right angle. 



E 
F 


D 


\ 


A < 


\ 6 



Proposition 12. 

To a given infinite straight line, from a given point 
which is not on it, to draw a perpendicular straight line. 

Let AB be the given infinite straight line, and C the 
given point which is not on it ; 



t 

I 




I. 12] PROPOSITIONS II, 12 271 

5 thus it is required to draw to the given infinite straight 
line AB^ from the given point 
C which is not on it, a per- 
pendicular straight line. 

For let a point D be taken 

10 at random on the other side of 

the straight line AB, and with 

centre c and distance CD let 

the circle EFG be described ; 

[Post 3] 
let the straight line EG 
IS be bisected at H, [i. 10] 

and let the straight lines CG, CH, CE be joined. 

[Post l] 

I say that CH has been drawn perpendicular to the given 
infinite straight line AB from the given point C which is 
not on it. 
20 For, since GH is equal to HE^ 
and HC is common, 

flie two sides GH, HC are equal to the two sides 
EH, HC respectively ; 
and the base CG is equal to the base CE ; 
25 therefore the angle CHG is equal to the angle EHC. 

[1.8] 

And they are adjacent angles. 

But, when a straight line set up on a straight line makes 

the adjacent angles equal to one another, each of the equal 

angles is right, and the straight line standing on the other is 

30 called a perpendicular to that on which it stands. [Def. 10] 

Therefore C/^ has been drawn perpendicular to the given 

infinite straight line AB from the given point C which is 

not on it 

Q. E. F. 

3. a perpendicular straight line, KdBrrm edtfciov ypaf»M^p, This is the full expression 
-for a perpiHdicular, ttiBrrw meaning lei faU or let dawn^ so that the expression corresponds 
to our plumb-line, ^ jrdtfrrot b however constantly used alone for a perpendicular, ypofifiilj 
being understood. 

ID. on the other side of the straight line AB, literally '* towards the other parts of 
the straight line AB,'' iwl rd Irfpa fU/ni r^t AB. Cf. **on the same side" (M rd o^d 
ftdfni) in Post. 5 and **in both directions" (^* ixdnpa rd fU/ni) in Def. 33. 

** This problem,*' says Produs (p. 283, 7—10), "was first investigated 
by Oenopides [5th cent b.c], who thought it useful for astronomy. He 
however calls the perpendicular, in the archaic manner^ (a line drawn) 



272 BOOK I [i. 12 

gmmum-wise (mra yr«fuyvaX because the gnomon is also at right anf^es to the 
horizon." In this earlier sense the gnomon was a staff {daoed in a vertical 
position for the purpose of casting shadows and so serving as a means of 
measuring time (Cantor, GesckickU der Mathimatik^ i,, p. 161). The later 
meanings of the word as used in Eucl. Book 11. and elsewhere will Le 
explained in the note on Book lu Def. 2. 

Proclus says that two kinds of perpendicular were distinguished^ the ''plane" 
(IrtircSof) and the "solid" (orfpco), the former being the perpendicular 
dropped on a line in a plane and the latter theperpenmcular dropped on a 
plane. The term "solid perpendicular" is sufficiently curious, but it may 
perhaps be compared with the Greek term ''solid locus" applied to a conic 
section, apparently on the ground that it has its origin in the section of a 
solid, namely a cone. 

Attention is called by most editors to the assumption in this proposition 
that, if only D be taken on the side oi AB remote from C, the circle described 
with CD as radius must necessarily cut A£ in two points. To sati^ us of 
this we need, as in i. i, some postulate of continuity, e.^. something like that 
suggested by Killing (see note on the Principle of Contmuity above, pu 235}: 
" If a point [here the point describing the arde] moves in a figure which is 
divided into two parts [by the straight line], and if it belongs at the beginning 
of the motion to one part and at another stage of the motion to the other 
ps^ it must during the motion cut the boundary between the two parts," and 
this of course applies to the motion in iufo directions fix>m D. 

But the editors have not, as a rul^ noticed a possible ctjecHon to the 
Euclidean statement of this problem which is much more difficult to dispose 
of at this stage, i.e. without employing any proposition later than this in 
Euclid's order. How do we know, says the supposed critic, that the cirde 
does not cut AB in thra or more points, in which case there would be not 
one perpendicular but three or more? Proclus (pp. 286, 12—289, 6) tries to 
refute this objection, and it is interesting to follow his argument, though it 
will easily be seen to be inconclusive. He takes in order three possible 
suppositions. 

I. May not the circle meet AB in a third point K between the middle 
point of GE and either extremity of it, taking the form drawn in the figure 
appended ? 

Suppose this possible. Bisect GE in H. Join CJ7, and produce it to 
• meet the circle in Z. Join CG^ CK, CE. 

Then, since CG is equal to CE^ and 
CH is common, while the base GH is 
equal to the base HE^ 

the angles CHG^ CHE are equal and, 
since they are adjacent, they are both right 

Again, since CG is equal to CE^ 
the angles at G and E are equal. 

Lastly, since CK is equal to CG and 
also to CE, the angles CGK, CKG are 
equal, as also are the angles CKE, CEK. 

Since the angles CGK, CEJCare equal, it follows that 

the angles CJCG, CITE are equal and therefore both right. 
Therefore the angle CKHis equal to the angle CHK, 
and CH\& equal to CK. 





I. 12] PROPOSITION 12 273 

But CK is equal to CZ, by the definition of the circle ; therefore CH is 
equal to CL : which is impossible. 

Thus Proclus; but why should not the circle meet AB m H2s well as JT? 

2. May not the circle meet AB in H the middle point of GE and take 
the form shown in the second fi^re? 

In that case, says Proclus, jom CG^ CHy CE as before. Then bisect HE 
at iT, join CK and produce it to meet 
the circumference at 2. 

Now, since HK is equal to KE^ CK 
is common, and the base CH is equal to 
the base CE^ 

the angles at K are equal and therefore 
both right angles. 

Therefore the angle CHK is equal to 
the angle CKH^ whence CKv& equal to CH 
and therefore to CL\ which is impossible. 

So Proclus ; but why should not the circle meet AB in JT as well as /T? 

3. May not the circle meet AB in two points besides G^ E and pass, 
between those two points, to the side of AB towards C, as in the next figure ? 

Here again, b^ the same method, Proclus proves that, K^ L being the 
other two points m which the circle cuts 
AB, 

CK is equal to CH, 
and, since the circle cuts CH in M, 

CM is equal to CK and therefore to 

tCH\ which is impossible. 
But, again, why should the circle not 
\ cut AB in the point Hz& well? 

In &ct, Proclus' cases are not mutually 
exclusive, and his method of proof only enables us to show that, if the circle 
meets AB in one more point besides G^ E^ it must meet it in more points 
still We can always find a new point of intersection by bisecting the distance 
separating any two points of intersection, and so, applying the method ad 
infinitum, we should have to conclude ultimately that the circle with radius 
CH (or CG) coincides with AB, It would follow that a drcle with centre 
C and radius greater than CH would not meet AB at all. Also, since all 
straight lines from C to points on AB would be equal in length, there would 
be an infinite number of perpendiculars from C on AB. 

Is this under any circumstances possible? It is not possible in Euclidean 
space, but it is possible, under the Riemann hypothesis (where a straight line 
is a "closed series" and returns on itself), in the case where C is the pole of 
the straight line AB. 

It is natural therefore that, for a proof that in Euclidean space there is 
only one perpendicular firom a point to a straight line, we have to wait until 
1. 16, the precise proposition which under the Riemann hypothesis is only valid 
with a certain restriction and not universally. There is no difficulty involved 
by waiting until i. 16, since i. 12 is not used before that proposition is reached ; 
and we are only in the same position as when, in order to satisfy ourselves of 
the number of possible solutions of i. i, we have to wait till i. 7. 
I But if we wish, after all, to prove the truth of the assumption without 
recourse to any later proposition than i. 12, we can do so by means of this 
same invaluable i. 7. 

H. E. 18 




«74 



BOOK I 




If the circle intersects AB as before in tr, jE^ let i7be the middle point of 
GE^ and suppose, if possible, that the 
circle also intersects AB in any other point 
AT on ^^. 

From H^ on the side of AB opposite to 
C, draw HL at right angles to AB^ and 
make HL equal to HC. 

Join CG, LG, CK, LK. 

Now, in the tnangles CBG^ LHG^ 
CH'xs equal to LH^ and HG is common. 

Also the angles CHG, LHG^ being 
both right, are equal. 

Therefore the base CG is equal to the base LG. 

Similarly we prove that CK is equal to LK. 

But, by hypothesis, since A* is on the circle, 
CK\& equal to CG. 

Therefore CG, CK, LG, LKzxe all equal. 

Now the next proposition, i. 13, will tell us that Clf, HL are in a straight 
line; but we will not assume this. Join CZ. 

Then on the same base CL and on the same side of it we have two pairs 
of straight lines drawn from C, L to G and K such that CG b equal to CK 
and LG to LK 

But this is impossible [i. 7]. 

Therefore the circle cannot cut BA or BA produced in any point other 
than G on that side of CL on which G is. 

Similarly it cannot cut AB or AB produced at any point other than E 
on the other side of CL> 

The only possibility left therefore is that the circle might cut AB in the 
same point as that in which CL cuts it But this is shown to be impossible 
by an adaptation of the proof of i. 7. 

For the assumption is that there may be some point M on CL such that 
CM is equal to CG and ZAT to LG. 

If possible, let this be the case, and produce CG 
toN. 

Then, since CM is equal to CG, 
the angle NGM is equal to the angle GML [i. 5, part 2]. 

Therefore the angle GML is greater than the angle 
MGL. 

Again, since LG is equal to LM, 
the angle GML is equal to the angle MGL. 

But it was also greater : which is impossible. 

Hence the circle in the original figure cannot cut AB in the point in 
which CL cuts it 

Therefore the circle cannot cut AB xn any point whatever except G and E. 

[This proof of course does not prove that CK is less than CG, but only 
that it is not equal to it The proposition that, of the obliques drawn 
from C to AB, that is Jess the foot of which is nearer to ZTcan only be proved 
later. The proof by i. 7 also fails, under the Riemann hypothesis, if C, Z are 
the poles of the straight line AB, since the broken lines CGL, CKL etc. 
become equal straight lines, all perpendicular to AB\ 

Produs rightly adds (p. 289, 18 sqq.) that it is not neassary to take D on 
the side of AB away from A^iwa objector ''says that there is no space on 





I. 12, 13] PROPOSITIONS 12, 13 275 

that side." If it is not desired to trespass on that side of AB^ we can take D 
anywhere on AB and describe the €irc of a circle between D and the point 
where it meets AB again, drawing the arc on the side of AB on which C is. 
If it should happen that the selected point D is such that the circle only meets 
AB in one pomt {D itself), we have only to describe the circle with CD as 
radius, then, if ^ be a point on this circle, take /*a point further from C than 
E is, and describe with CF as radius the circular arc meeting AB in two 
points. 



Proposition 13. 

If a straight line set up an a straight line make angles^ it 
will make either two right angles or angles equal to two right 
angles. 

\ For let any straight line AB set up on the straight line 

s CD make the angles CBA, ABD ; 

I say that the angles CBA, ABD 
are either two right angles or equal to 
two right angles. 

Now, if the angle CBA is equal to g 

to the angle ABD, 

they are two right angles. [Def. 10] 

But, if not, let BE be drawn from the point B at right 

' angles to CD\ [i. n] 

therefore the angles CBE, EBD are two right angles, 
fis Then, since the angle CBE is equal to the two angles 
CBA, ABE, 

let the angle EBD be added to each ; 
therefore the angles CBE, EBD are equal to the three 
angles CBA, ABE, EBD. , [C. N. 2] 

Again, since the angle DBA is equal to the two angles 
DBE, EBA, 

let the angle ABC be added to each ; 
therefore the angles DBA, ABC are equal to the three 
angles DBE, EBA, ABC. [C N. 2] 

i But the angles CBE, EBD were also proved equal to 
the same three angles ; 

and things which are equal to the same thing are also 
equal to one another ; [C. N. 1] 

therefore the angles CBE, EBD are also equal to the 
.0 angles DBA, ABC 

18—2 



276 BOOK I [h 13, 14 

But the angles CBB, EBD are two right angles ; 

therefore the angles DBA, ABC are also equal to two 

right angles. 

Therefore etc. 

Q. E. D. 

17. let the angle EBD be added to each, literally ** let the angle EBD be added 
(so as to be) common,'* K9Uf^ wfnontiffBu ^ W6 BBA. Sinularly irouH^ d^g^^ is med of 
subtracting a straight line or angle from each of two others. "Let the common angle EBD 
be added '^is clearly an inaccurate translation, for the angle is not common before it is added, 
i.e. the irouH^ is proleptic. ** Let the common angle be sttkraOad** as a translation of atu^ 
d^pf^Bw woula be less unsatisfactory, it is true, out, as it is desirable to use correraondhig 
words when translating the two expressions, it seems hopeless to attempt to keep toe won 
« common,'* and I have therefore said ** to each " and " nrom each " simpfy. 



Proposition 14. 

If with any straight line, and at a point on it, two stra^kt 
lines not lying on the same side make the adjacent angles equal 
to two right angles, the two straight lines will be in a straight 
line with one another. 

5 For with any straight line AB, and at the point B on it, 
let the two straight lines BC, BD not lying on the same side 
make the adjacent angles ABC, ABD equal to two right 
angles ; 

I say that BD is in a straight line with CB. 
o For, if BD is not in a straight line 
with BC, let BE be in a straight line 
with CB. 

Then, since the straight line AB g — 
stands on the straight line CBE, 
s the angles ABC, ABE are equal to two right angles. 

[1. 13] 
But the angles ABC, ABD are also equal to two right angles ; 
therefore the angles CBA, ABE are equal to the angles 
CBA, ABD. [Post 4 and C. N. i] 

Let the angle CBA be subtracted frpm each ; 
» therefore the remaining angle ABE is equal to the remaining 
angle ABD, [C N. 3] 

the less to the greater : which is impossible. 
Therefore BE is not in a straight line with CB. 
Similarly we can prove that neither is any other straight 
\s line except BD. 




I. 14, is] propositions 13—15 277 

Therefore CB is in a straight line with BD. 
Therefore etc. ^ « ^ 

Q. E. D. 

I. If with any straight line.... There is no greater difficulty in translating the works 
of the Greek geometers than that of accurately giving the force of prepositions, rp^t, for 
Instance, is used in all sorts of expressions with various shades of meaning. The present 
enunciation begins *E^ irpbt ripi tidel^ koI rf *-p6t a&ri ffri/^ff% *nd it is rdly necessary in 
this one sentence to translate rpdf by three different words, witAt at, and am. The first wp^ 
must be translated by wit A because two straight lines " makj" an angle vniA one another. On 
the other hand, where the similar expression wpits rj BodelffTg tdBtl^ occurs in 1. 33, but it is 
a question of "constructing" an angle (o^uon^ao'tfac), we have to sav "to construct ch a 
given straight line." Against would perhaps be the English word comine nearest to 
expressing all these meanings of «7>6f, but it would be intolerable as a tranuation. 

17. Todhunter points out that tor the inference in this line Post. 4, that all right angles 
are equal, is necessary as well as the Common Notion that things which are equal to the same 
thing (or rather, here, to e^uat tAings) are equal. A similar remark applies to steps in the 
pro(»s of 1. 15 and i. 38. 

34. we can prove. The Greek expresses this by the future of the verb, de(^>/ier, 
** we shall prove,'* which however would perhaps be misleading in English. 

Proclus observes (p. 297) that two straight lines on the same side of another 
straight line and meeting it in one and the same 
point may make with one and the same portion 
of the straight line terminated at the pomt two 
angles which are together equal to two right angles, 
in which case however the two straight lines would 
not be in a straight line with one another. And 
he quotes from Porphyry a construction for two 
such straight lines in the particular case where they 
form with the given straight line angles equd 
respectively to half a right angle and one and a 
half right angles. There is no particular value in 
the construction, which will be gathered from the annexed figure where CE^ 
CF are drawn at the prescribed inclinations to CD. 



Proposition 15- 

If two straight lines cut one another, they make the vertical 
angles equal to one another. 

For let the straight lines AB, CD cut one another at the 
point E ; 

I say that the angle A EC is equal to ^.^^ 

the angle DEB, ^^>-^ 

and the angle CEB to the angle o ^^<~c 

AED. § 

For, since the straight line AE stands 
3 on the straight line CD, making the angles CEA, AED, 
the angles CEA, AED are equal to two right angles. 

[I- 13] 




378 BOOK I [l 15 

Again, since the straight line DE stands on the straight 
line AB, making the angles AED\ DEB, 

the angles AED, DEB are equal to two right angles. 

IS But the angles CEA, AED were also proved equal to 
two right angles ; 

therefore the angles CEA, AED are equal to the 
angles AED, DEB. * [Post 4 and C A: i] 

Let the angle AED'h^ subtracted from each ; 
» therefore the remaining angle CEA is equal to the 
remaining angle BED. [C. N. 3] 

Similarly it can be proved that the angles CEB^ DEA 
are also equal. 

Therefore etc. q. E. d. 

25 [PoRiSM. From this it is manifest that, if two straight 
lines cut one another, they will make the angles at the point 
of section equal to four right angles.] 

I. Uie vertical angles. The diiference between atffacmt angles («l tf^cf^t T wwht) and 
vertuai angles {ol irarA Kofio^ >Mr(ai) is thus explained by Produs {p, 198, 14—14). The 
first term describes the angles made by two straight lines whfXk one onlv is dividedf by the 
other, i.e. when one straight line meets another at a point which is not either of its extremi- 
ties, but is not itself produced beyond the point of meeting. When the first straight line is 
produced, so that the lines cross at the pomt, they make two pairs of vertuai angles (whidi 
are more clearly described as vertically opposite angles), and which are so called because their 
converrcnce is from opposite directions to one point (the intersection of the lines) as vertex 
(«»pi^?). 

36. at the point of section, literally "at the section," rpftt r% ro/ii. 

This theorem, according to Eudemus, was first discovered by Thales, but 
found its scientific demonstration in Euclid (Proclus, p. 299, 3—^). 

Proclus gives a converse theorem which may be stated thus. If a straight 
line is met at one and the same point intermediate in its length by two other 
straight lines on different sides of it and such as to mahe the vertical angles 
equals the latter straight lines are in a straight line with one another. The 
proof need not be given, since it is almost self-evident, whether (i) it is dirtet, 
by means of i. 13, 14, or (2) indirect, by reductio ad absurdum depending 
on I. 15. 

The balance of MS. authority seems to be against the genuineness of this' 
Porism, but Proclus and Psellus both have it. The word is not here used, as it 
is in the title of Euclid's lost Porisms, to si^ify a particular class of independent 
propositions which Proclus describes as bemgm some sort intermediate oetween 
theorems and problems (requiring us, not to bring a thing into existence, but 
to^iM/somethmg which we know to exist). Porism has here (and wherever 
the term is used in the Elements) its second meaning ; it is what we call a 
corollary, i.e. an incidental result springing from the proof of a theorem or the 
solution of a problem, a result not directly sought but appearing as it were by 
chance without any additional labour, and constituting;, as Produs sa3rs, a sort 
of windfall (^/acuov) and bonus (ic^os). These Ponsms appear in both the 



I. 15, 16] PROPOSITIONS IS, 16 279 

geometrical and arithmetical Books of the Elements^ and may either result 
from theorems or problems. Here the Porism is geometrical, and springs out 
of a theorem ; vii. 2 affords an instance of an arithmetical Porism. As an 
instance of a Porism to a problem Proclus cites " that which is found in the 
second Book'' (ro iv rf Scvr^ fttpkuf ccifuvov) ; but as to this see notes on 

II. 4 and IV. 15. 

The present Porism, says Produs, formed the basis of ''that paradoxical 
theorem which proves that only the following three (regular) polygons can fill 
up the whole space surrounding one point, the equilateral triangle, the square, 
and the equilateral and equiangular hexagon." We can in fact place round a 
point in this manner six equilateral triangles, three regular hexagons, or four 
squares. "But only the angles of these regular figures, to the number specified, 
can make up four right angles : a theorem due to the Pythagoreans." 

Proclus further adds that it results from the Porism that, if any number of 
straight lines intersect one another at one point, the sum of all the angles so 
formed will still be equal to four right angles. This is of course what is 
. generally given in the text-books as Corollaiy 2. 

Proposition 16. 

In any triangle^ if one of the sides be produced^ the exterior 
angle is greaier than either of the interior and opposite angles. 
Let ABC be a triangle, and let one side of it BC be 
produced to D\ 
5 I say that the exterior angle ACD is greater than either 
of the interior and opposite angles 
CBA, BAC. 

Let A Che bisected at £ [i. lo], 
and let B£ be joined and produced 
10 in a straight line to £] 

let £Fhe made equal to B£[i. 3], 
let FC be joined [Post, i], and let A C 
be drawn through to G [Post. 2]. 
Then, since A£ is equal to £Ct 
cs and B£ to £F, 

the two sides A£, £B are equal to the two sides C£, 
£F respectively ; 

and the angle A£B is equal to the angle F£Cy 

for they are vertical angles. [i. 15] 

JO Therefore the base AB is equal to the base FC, 

and the triangle AB£ is equal to the triangle CF£, 
and the remaining angles are equal to the remaining angles 
, respectively, namely those which the equal sides subtend ; [i. 4J 
therefore the angle BAB is equal to the angle ECF. 




a8o BOOK I [l i6 

; But the angle ECD is greater than the angle ECF\ 

therefore the angle ACD is greater than the angle BAE. 

Similarly also, if BC be bisected, the angle BCG, that is, 
the angle ACD [i. 15], can be proved greater than the angle 
ABC as well. 

Therefore etc. Q. E. d. 

1. the exterior angle, literally ** the ontside angle,** ^ ifnh^ ymUa. 

2. the interior and opposite angles, rOir irr^ jral dhrtroyrlar 7««Mi)ir. 
II. let AC be drawn through to O. The word is &4x^«#, a variatioo on the more 

usual iKPifiXili^ew, "let it ht producetL'* 
II. CFE, in the text •• FEC' 

As is well known, this proposition is not universally tnie under the 
Riemann hypothesis of a space endless in extent but not infinite in size. On 
this hypothesis a straight line is a *' closed series" and returns on itself; and 
two straight lines which have one point of intersection have another point of 
intersection also, which bisects the whole lengdi of the straight line measured 
from the first point on it to the same point again; thus the axiom of Euclidean 
geometry that two straight lines do not enclose a space does not hold. If 4A 
denotes the finite length of a straight line measured from any point once 
round to the same point again, a A is the distance between the two intersecticHis 
of two straight lines which meet Two points A^ B do not determine one 
sole straight line unless the distance between them is diffioent firom 2A. ^ In 
order that there may only be one perpendicular fix>m a point C to a straight 
line AB^ C must not be one of the two " poles " of the straight line. 

Now, in order that the proof of the present proposition may be universally 
valid, it is necessary that C/* should always fall within the angle ACD so that 
the angle ACFmay be less than the angle ACD. But this will not always be 
so on the Riemann hypothesis. For, (i) if BE is equal to A, so that BF is 
equal to 2 A, FmW be the second point in which BE and BD intersect ; i.e. 
/•will lie on CD, and the angle ACF will be egua/ to the angle ACD. In 
this case the exterior angle ACD will be e^al to the interior angle BAC> ^ 
(2) If BE is greater than A and less than 2A, so that BF is greater than 2A 1 
and less than 4A, the angle ACF will be greater than the angle ACD, and | 
therefore the angle ACD will be less than the interior angle BAC. Thus, e.g.y | 
in the particular case of a right-angled triangle, the angles other than the right j 
angle may be (i) both acute, (2) one acute and one obtuse, or (3) both obtuse 
according as the perpendicular sides are (i) both less than A, (2) one less and 
the other greater than A, (3) both greater than A. 

Proclus tells us (p. 307, 1 — 12) that some combined this theorem with the 
next in one enunciation thus: In any triangle, if one side be produced, the 
exterior angle of the triangle is greater than either of the interior and opposite 
angles, and any two of the interior angles are less than two right angles, the 
combination having been suggested by the similar enunciation of EucUd l 32, 
In any trian^, if one of the sides be produced, the exterior angle is equal to the 
two interior and opposite angles, and the three interior angles of the trian^ are 
equal to two right angUs. 

The present proposition enables Proclus to prove what he did not succeed 
in estabhshing conclusively in his note on 1. 12/ namely ^BcaX from one point 
there cannot be drawn to the same straight line three straight lines equal in UngtK 



1. i6, 17] 



PROPOSITIONS 16, 17 



381 



I 



let AB, AC, AD be all equal, £, C, D being in a 
equal, the angles 




For, if possible, 
straight line. 

Then, since AB, AC are 

ABC, ACB are equal. 
Similarly, since AB, AD are equal, the angles 

ABD, ADB are equal. 
Therefore the angle ACB is equal to the angle 

A DC, i.& the exterior angle to the interior and 
opposite angle: which is impossible. 

Proclus next (p. 308, 14 sqq.) undertakes to prove by means of i. 16 that, 
if a straight line falling on two straight lines make the exterior angle equal to 
the interior and opposite angle, the tufO straight lines will not form a triangle or 
meet, for in that case the same angle would be both greater and equal. 

The proof is really equivalent to that of EucL i. 27. V BE falls on the 
two straight lines AB, CD in such a wa^ that the angle 
CDE is equal to the interior and opposite angle ABD, 
AB and CD cannot form a triangle or meet. For, if 
they did, then (by i. 16) the angle CDE would be 
greater than the an^le ABD, while by the hypothesis 
it is at the same time equal to it 

Hence, says Proclus, in order that BA^ DC may 
form a triangle it is necessary for them to approach one 
another in the sense of being turned round one pair of 
corresponding extremities, e.g. B, D, so that the other extremities A, C come 
nearer. This may be brought about in one of three ways: (i) AB may 
remain fixed and CD be turned about D so that the angle CDE increases ; 
(2) CD may remain fixed and AB be turned about B so that the angle ABD 
becomes smaller; (3) both AB and CD may move so as to make die angle 
ABD smaller and the angle CDE larger at the same time. The reason, then, 
of the straight lines AB, CD coming to form a triangle or to meet is (says 
Proclus) the movement of the straight lines. 

Though he does not mention it here, Proclus does in another passage 
(p. 371, 2 — 10, quoted on p. 207 above) hint at the possibiUty that, while i. 16 
may remain universally true, either of the straight ?ixit& BA, DC (or both 
together) may be turned through any angle not greater than a certain finite 
angle and yet may not meet (the Bolyai-Lobachewsky hypothesis). 



r 



Proposition 17. 

In any triangle two angles taken together in any manner 
-are less than two right angles. 
Let ABC be a triangle ; 

I say that two angles of the triangle ABC taken together in 

any manner are less than two right angles. 

For let BC be produced to I). [Post. 2] 

Then, since the angle A CD is an exterior angle of the 

triangle ABC, 

it is greater than the interior and opposite angle ABC 

[1. 16] 



282 BOOK I [l 17 

Let the angle ACB be added to each ; 
therefore the angles A CD, ACB are greater than the angles 
ABC, BCA. 




But the angles A CD, ^C^^raj equal to two right angles. 

[I. «3] 



Therefore the angles ^/^|^ BCA are less than two right 
imilarly we can prove thitoythe angles BAC, ACB are 



angles. f 

we can prove uiiMyuic au^ica ajxt,\^, xt,\^aj mc 

th an Ji^o right^anglc^and so are the angles CAB, 

AiclSm^.,^ pi %. 

Thereiore>6tc. ">'sk*.,.^ 

I ^^^^""^^^^ ^* ^' ^' 

I. taken Sogether in any manner, vd^rT^MraXaM/fftv^^MFai, Le. any pair added 
togetlier. V 

As in his note on the previous propodtiony Produs tries to state the comu 
of the property. He takes the case of two straight lines forming right anises 
with a transversal and observes that it is the convergence of the straight lifus 
towards one another (irvvcvo'if rwf cv^coSy), the lessening oi the two right angles, 
which produces the triangle. He will not have it that the &ct of the exterior 
angle being greater than the interior and opposite angle is the cause of the 
property, for the odd reason that '' it is not necessary that a side should be 
produced, or that there should be any exterior angle constructed... and how can 
what is not necessary be the cause of what is necessary?" (p. 311, 17 — 21). 

Agreeably to this view, Proclus then sets himself to prove the theorem 
without producing a side of the triangle. 

Let ABC be a triangle. Take any point D on 
BC, and join AD. 

Then the exterior angle ADC of the triangle ABD 
is greater than the interior and opposite angle ABD. 

Similarly the exterior angle ADB of the triangle 
ADC is greater than'tlie interior and opposite angle 
ACD. 

Therefore, by addition, the angles ADB, ADC are together greater than 
the angles ABC, ACB. 

But the angles ADB, ADC are equal to two right angles ; therefore the 
angles ABC, ACB are less than two right angles. - 

Lastly, Proclus proves (what is obvious from this proposition) that there 
cannot be more than one perpendicular to a straight line from a point without 
it. For, if this were possible, two of such perpendiculars would form a triangle 
in which two angles would be right angles: which is impossible, since any two 
angles of a triangle are together less tbuui two right angles. 




I. i8] 



PROPOSITIONS 17, 18 

Proposition 18. 



283 




In atiy triangle the greater side subtends the greater angle. 

For let ABC be a triangle having the side AC greater 
than AB ; 

I say that the angle ABC is also greater than the angle 
BCA. 

For, since AC\^ greater than AB, let AD be made equal 
to AB [i. 3], and let BD be joined. 

Then, since the angle ADB 
is an exterior angle of the triangle 
BCD, 

it is greater than the interior 
and opposite angle DCB. [l 16] 

But the angle ADB is equal 
to the angle ABD, 

since the side AB is equal to AD ; 

therefore the angle ABD is also greater than the angle 
ACB] 

therefore the angle ABC is much greater than the angle 
ACB. 

Therefore etc. 

Q. E. D. 

In the enunciation of this proposition we have ^orc/yecv (''subtend 'O used with the 
simple accusative instead of the more usual inrh with accusative. The latter construction 
is used in the enunciation of i. 19, which otherwise only differs from that of 1. 18 in the order 
of the words. The point to remember in order to distinguish the two is that the daium 
comes first and the quauitum second, the datum being in this proposition the greater side 
and in the next the greater angie. Thus the enunciations are (l. 18) wwnht rpiyiiifov if /i/d{taif 
wXtvpik rifif fktl^wa yw^lnM intvrthti and (l. 19) waanht rpvyfiifw ^6 rV jul^a ytfifUuf ^ 
tMtl{m irXcvpd ^or€h€t. In order to keep the proper order in En^lidi we must use the 
passive of the verb in i. 19. Aristotle quotes the result of i. 19, using the exact wording, 
ord Tdp "Hiw fuli^ ytfiflaw inrorc^ci (Afeteorologica III. 5, 376 a is). 

''In order to assist the student in remembering which of these two 
propositions [i. 18, 19] is demonstrated directly and 
'" which indirectly, it may be observed that the order is 
similar to that in i. 5 and i. 6" (Todhunter). 

An alternative proof of i. 18 given by Porphjny 
(see Proclus, pp. 315, 11 — 316, 13) is interesting. It 
starts by supposing a length equal to A£ cut off from 
the other end of -^C; that is, CD and not AD is 
made equal to AB. 

Produce AB to E so that BE is equal to AD, and 
join EC 

Then, since AB is equal to CD, and BE to AD, 
AE is equal to AC 





384 BOOK I [l 18, 19 

Therefore the angle ARC is equal to the angle ACE. 

Now the angle ABC is greater than the angte AEC^ [l 16] 

and therefore greater than the angle ACE. 
Hence, a fortiori^ the angle ABC is greater than the angle ACB. 

Proposition 19. 

In any triangle the greater angle is subtended by the 
greater side. 

Let ABC be a triangle having the angle ABC greater 
than the angle BCA ; 

I say that the side AC is also greater than the side AB. 

For, if not, -^C is either eaual to AB or less. 

Now -^C is not equal to AB ; 
for then the anc^le ABC would also have been 
equal to the angle ACB ; [i. sj 

but it is not ; 

therefore AC is not equal to AB. 

Neither is -^C less than AB, 
for then the angle ABC would also have been less than the 
Single ACB; [1. iS] 

but it is not ; 

therefore AC is not less than AB. 

And it was proved that it is not equal either. 
Therefore -^C is greater than AB. 

Therefore etc. Q. e. d. 

This proposition, like i. 6, can be proved by merely /dguai deduction from 
I. 5 and I. 18 taken together, as pointed out by De Morgan. The seneral 
form of the argument u^ by De Morgan is given in his Format Zcgic {iS4j\ 
p. 25, thus : 

^^ Hypothesis. Let there be any number of propositions or assertions — 
three for instance, X, Y and Z— of which it is tiie property that one or the 
other must be true, and one only. Let there be three other propositions 
Py Q and ^ of which it is also the property that one, and one only, must be 
true. Let it be a connexion of those assertions that : 

when X is true, P is true, 

when yis true, Q is true, 

when Z is true, ^ is true. 
Consequence: then it follows that, 

when P)S true, X is true, 

•when Q is true, K is true, 

when R is true, Zis true." 






I 

I 



[1 


5] 


[I- 


i8] 


[' 


.6] 


[I- 


^9] 



I. 19] PROPOSITIONS 18, 19 385 

To apply this to the case before us, let us denote the sides of the triangle 
ABC by tf, by r, and the angles opposite to these sides by A^ B^ C respectively, 
and suppose that a is the base. 

Then we have the three propositions, 

when b is equal to r, ^ is equal to C, 
when b is greater than r, B is greater than C, 1 
when b is less than r, B is less than C, / 

and it follows lagUally that, 

when B is equal to C, b is equal to r, 
\ when B is greater than C, ^ is greater than r, 1 

when B is less than C, ^ is less than e. / 

Reductio ad absurdum by exhaustion.- 

Here, says Proclus (p. 318, 16 — 23), Euclid proves the impossibili^ "by 
means of division^* {U huupkr^ioi). This means simply the separation of 
different hypotheses, each of which is inconsistent with the truth of the 
theorem to be provcKl, and which therefore must be successively shown to be 
impossible. If a straight line is not greater than a straight line, it must be 
either equal to it or less ; thus in a reductio ad absurdum intended to prove 
such a theorem as i. 19 it is necessary to dispose successively of Hue hypotheses 
inconsistent with the truth of the theorem. 

Alternative (direct) proof. 

Proclus gives a direct proof (pp. 319 — 321) which an-Nairia also has and 
attributes to Heron. It requires a lemma and is consequently open to the 
slight objection of separating a theorem from its converse. But the lemma 
and proof are worth giving. 

Lemma. 

Jfan angle of a triangle be bisected and the straight line bisecting it meet the 
base and dMde it into unequal parts^ the sides containing the angle will be 
uncqucUy and the greater will be that which meets the greater segment of the base^ 
and the las thcU which meets the less. 

Let ADy the bisector of the angle A of the triangle ABC^ meet BC in Z>, 
making CD greater than BD, 

I say that AC is greater than AB. 
Produce AD ioE so that DE is equal to 
AD. And, since Z>C is greater than BD^ cut 
oflDFeqvaltoBD. 

Join EFand produce it to G. 
Then, since the two sides AD, DB are 
equal to the two sides ED^ DF^ and the 
vertical angles at D are equal, 

AB is equal to EF^ 
and the angle DEF io the angle BAD, 

i.e. to the angle DAG (by hypothesis). 
Therefore AG\s equal to EG^ [1. 6] 

and therefore greater than EF, or AB. 
Hence, afartiori, AC\& greater than AB. 



U 





286 BOOK I [i. 19, to 

Proof of I. zg. 

Let ABC be a triangle in which the angle ABC is greater than the ang^ 
ACB* 

Bisect BC at Z>, join AD^ and produce it to j? so that DE is equal to 
AD, ]o\ti BE. 

Then the two sides BD^ DE are equal to the two 
sides CD^ DA^ and the vertical angles at Z> are equal ; 

therefore BE is equal to AC^ 

and the angle DBE to the angle at C 

But the angle at C is less than the angle ABC\ 

therefore the angle DBE is less than the angle 
ABD. 

Hence, if BF bisect the angle ABE^ BF meets 
AE between A and D. Therefore EF is greater 
than FA. 

It follows, by the lemma, that BE is greater than 
BA, 

that is, AC is greater than AB. 



Proposition 20. 

In any triangle two sides taken together in any manner 
are greater than the remaining one. 

For let ABC be a triangle ; 
I say that in the triangle ABC two sides taken together in 
any manner are greater than the remaining one, namely 
BAy AC greater than BC, 
AB, BC greater than AC, 
BCy CA greater than AB. 
For let BA be drawn through to the point /?, 
let DA be made equal to CA, and let DC be 
joined. 

Then, since DA is equal to AC, 

the angle ADC is also equal to the angle 

ACD; [I.S] 

therefore the angle BCD is greater than 

the angle ADC. [C N. 5] 

And, since DCB is a triangle having the angle BCD 
greater than the angle BDC, 

and the greater angle is subtended by the greater side, 

[i. 19] 
therefore DB is greater than BC. 




I. 20] PROPOSITIONS 19, ao 287 

But DA is equal to AC; 

therefore BA, AC are greater than BC. 
Similarly we can prove that AB, BC are also greater 
than CA, and BC, CA than AB. 
Therefore etc. 

Q. E, D. 

It was the habit of the Epicureans, says Proclus (p. 323), to ridicule this 
theorem as being evident even to an ass and requiring no proof, and their 
all^ation that the theorem was ''known" (yvwpifiov) even to an ass was based 
on the fact that, if fodder is placed at one angular point and the ass at another, 
he does not, in order to get to his food, traverse the two sides of the triangle 
but only the one side separating them (an argument which makes Savile exclaim 
that its authors were "digni ipsi, qui cum Asino foenum essent," p. 78). 
Proclus replies truly that a mere perception of the truth of the theorem is a 
different thing from a scientific proof of it and a knowledge of the reason why 
it is true. Moreover, as Simson says, the number of axioms should not be 
increased without necessity. 

Alternative Proofs. 

Heron and Porphyry, we are told (Proclus, pp. 323 — 6), proved this 
tfieorem in different ways as follows, without producing one of the sides. 

J^rs/ proof. 

Let ABC be the triangle, and let it be required to prove that the sides 
BA, AC are greater than BC 

Bisect the angle BA C by AD meeting BC in D. 

Then, in the triangle ABD, 
. the exterior angle ADC is greater than the 
interior and opposite angle BAD, [i. 16] 

that is, greater than the angle DAC 

Therefore the side ^C is greater than the side 
CD. [i. 19] 

Similarly we can prove that AB is greater than BD, 

Hence, by addition, BA, ACsoregjreaittr than BC. 

Second proof . 

This, like the first proof, is direct. There are several cases to be considered. 

ii) If the triangle is equilateral, the truth of the proposition is obvious. 
2) If the triangle is isosceles, the proposition needs no proof in the case 
(a) where each of the equal sides is ^eater than the base. 

(b) If the base is greater than either of the other sides, we have to prove 
that the sum of the two equal sides is greater than 
the base. Let BC be the base in such a triangle. 

Cut off from BC a length BD equal to AB, and 
join AD. 

Then, in the triangle ADB, the exterior angle 
ADC is greater than the interior and opposite angle 
BAD. • [i. 16I 

Similarly, in the triangle ADC, the exterior angle ADB is greater than the 
interior and opposite angle CAD. 





388 BOOK I [l 3o 

By addition, the two angles BDA^ ADC are together greater than the 
two angles BAD, DAC (or the whole angle BAC). 

Subtracting the equal angles BDA, BAD^ we have tfie angle ADC 
greater than the angle CAD. 

It follows that ^C is greater than CD\ [i. 19] 

and, adding the equals AB^ BD respectively, we have BA^ AC together 
greater than BC 

(3) If the triangle be scakm^ we can arrange the sides in order of lensth. 
Suppose BC is the greatest, AB the intermediate and AC the least side. 
Then it is obvious that AB, BC are together greater than AC^ and BC^ CA 
together greater than AB, 

It only remains therefore to prove that CA, AB are together greater 
than BC. 

We cut off from BC a length BD equal to the adjacent side^ join AD^ and 
proceed exactly as in the above case of the isosceles trian^e. 

Thirdpraof. 

This proof is by rtductio ad absurdum. 

Suppose that BC is the greatest side and, as before, we have to prove that 
BA, ACsLre greater than BC. 

If they are not, they must be either equal to 
or less than BC 





(i) Suppose BA, AC are together equal 
to BC. 

From BC cut off BD equal to BA, and 
join AD. 

It follows from the hypothesis that DC is equal to AC. 

Then, since BA is equal to BD, 
the angle BDA is equal to the angle BAD. 

Similarly, since ^C is equal to CD, 
the angle CDA is equal to the angle CAD. 

By addition, the angles BDA, ADC are together equal to the whole angle 
BAC. 

That is, the angle BAC is equal to two right angles : which is impossible. 

(2) Suppose BA, AC are together less than BC. 

From BC cut off BD equal to BA, and from CB cut off CE equal to 
CA. Join AD, AB. 

In this case, we prove in the same way that 
the angle BDA is equal to the angle BAD, and 
the angle CEA to the angle CAE. 

By addition, the sum of the angles BDA, 
AEC is equal to the sum of the angles BAD, 
CAE. 

Now, by I. 16, the angle BDA is greater than the angle DAC, and 
theiefore, a fariiari, greater than the angle EAC. 

Similarly the angle AEC is greater than the angle BAD. S 

Hence the sum of the angles BDA, AEC is greater than the sum of the ' 
angles BAD, EAC. 

But the former sum was also equal to the lAtter : which is impossible. 




! I. ai] PROPOSITIONS 20, 21 389 

Proposition 21. 

If on one of the sides of a triangle, from its extremities, 

there be constructed two straight lines meeting within the 

triangle, the straight lines so constructed will be less than the 

remaining two sides of the triangle, but will contain a grecUer 

I angle. 

j On BC, one of the sides of the triangle ABC, from its 

extremities B, C, let the two straight lines BD, DC be con- 
I structed meeting within the triangle ; 

I I say that BD, DC are less than the remaining^ two sides 

MO of the triangle BA, AC, but contain an angle BvC greater 
I than the angle BAC 

For let BD be drawn through to E. 

Then, since in any triangle two 

, sides are greater than the remaining 

ii5 one, [i. 20] 

therefore, in the triangle ABE, the 

\ two sides -^jff, AE are greater than BE. 

Let EC be added to each ; 

therefore BA, AC are greater than BE, EC 
|ao Again, since, in the triangle CED, 
\ the two sides CE^ ED are greater than CD, 

\ let DB be added to each ; 

therefore CE, EB are greater than CD, DB. 
But BA^ AC were proved greater than BE, EC; 

therefore BA, AC are much greater than BD, DC. 
Agrain, since in any triangle the exterior angle is greater 
than the interior and opposite angle, [i. 16] 

therefore, in the triangle CDE, 

the exterior angle BDC is greater than the angle CED. 
o For the same reason, moreover, in the triangle ABE also, 
the exterior angle CEB is greater than the angle BAC. 
But the angle BDCvias proved greater than the angle CEB; 
therefore the angle BDC is much greater than the angle 
BAC. 
»^ Therefore etc. q. e. d. 

3. be constructed... meeting within the triangle. The word "meeting" is not in 
the Greek, where the words are irrhs cveraBCMtv, ffwlffroffSai is the word lued of con- 
structing two straight lines to a point (cf. i. 7) or^ as to form a triangle ; but it is necessaiy 
in Engksh to indicate that they nuet. 

3. the straight lines so constructed. Observe the elegant brevity of the Greek ol 

H. B. 19 



«90 



BOOK I 



[i. ti 



1 




The editors generally call attention to the fiiurt that the lines dnwn within 
the triangle in this proposition must be drawn, 
as the enunciation says, from the ends of the 
side ; otherwise it is not necessary that their 
sum should be less than that of the remaining 
sides of the triangle. Proclus (p. 337, 1 2 sqq.) 
gives a simple illustration. 

Let i^^C be a right-angled triangle. Take 
any point D on BC^ join DA^ and cut off 
from it DE equal to AB, Bisect AE at A 
and join FC. 

Then shall CF, FD be together greater than CA^ AB. 
For CF, FE are equal to CF, FA, 
and therefore greater than CA. 

Add the equals ED, AB respectively ; 
therefore CF, FD are together greater than CA, AB. 

Pappus gives the same proposition as that just proved, but follows it up 
by a number of others more elaborate in character, selected apparently from 
" the socalled paradoxes " of one Erycinus (Pappus, in. p. 106 sqq.). Thus 
he proves the following : 

1. In any triangle, except an equilateral triangle or an isosceles triangle 
with base less than one of the other sido, it is possible to construct on the 
base and within the triangle two straight lines the sum of which is equal to 
the sum of the other two sides of the triangle. 

2. In any triangle in which it is possible to construct two straight lines on 
the base which are equal to the sum of the other two sides of the triangle it is 
also possible to construct two others the sum of which isgnaierXbBn that sum. 

3. Under the same conditions, if the base is greater than either of the 
other two sides, two straight lines can be constructed in the manner described 
which are respectively greater than the other two sides of the triangle ; and the 
lines may be constructed so as to be respectively equal to the two sides, if one 
of those two sides is less than the other and each of them less than the base. 

4. The lines may be so constructed that their sum wiU bear to the sum 
of the two sides of the triangle any ratio less than 2:1. 

As a specimen of the proofs we will give that of the proposition whidi has 




been numbered (i) for the case where the triangle is isosceles (Pappus, iii. 
pp. 108 — no). 



I. 2i] PROPOSITION 21 391 

Let ABC he an isosceles triangle in which the base ^C is greater than 
either of the equal sides AB, BC. 

With centre A and radius AB describe a circle meeting ACin D. 

Draw any radius AEJ^such that it meets ^Cin a point jF outside the circle. 

Take any point G on EFy and through it draw (7^ parallel to AC. Take 
any point JTon GJfy and draw J^L parallel to IrA meetmg i^C in Z. 

From BC cut off -5-A^ equal to EG. 

Thus ^^, or LJiiy is equal to the sum of AB^ BN^ and CTV^is less than LK, 

Now GFy FHzxe together greater than GH, 
and CH, /T^ together greater than CK. 

Therefore, by addition, 
CA FGy BK9je together greater than CK, HG. 

Subtracting HK from each side, we see that 
CF, FG are together greater than CK, KG \ 
therefore, if we add AG to each, 

AF, FC are together greater than AG, GK, KC 

And AB, BCsLTt together greater than AF, FC. [i. 21] 

Therefore AB, BCaie together greater than AG, GK, KC 

But, by construction, AB, BN dse together equal to ^^; 
therefore, by subtraction, NC is greater than GK, KC, 
and a fortiori greater than KC. 

Take on KC produced a point Jlfsuch that KM is equal to NC; 
with centre A!* and radius JTJl/' describe a circle meeting CL in O, and join KO. 

Then shall LK, KO be equal to AB, BC. 

For, by construction, ZK is equal to the sum of AB, BN, and KO is 
equal to NC\ 

therefore LK, KO sure together equal to AB, BC 

It is after i. 31 that (as remarked by De Morgan) the important 
proposition about the perpendicular and obliques drawn from a point to a 
straight line of unlimited length is best introduced : 

0/ all straight lines that can be drawn to a given straight line of unlimited 
length from a given point without it: 

(a) the perpendicular is the shortest; 

(d) of the obliques, that is the greater the foot of which is further from the 
perpendicular ; 

{c) given one oblique, only one other can be found of the same length, namely 
that the foot of which is equally distant with the foot of the given one from the 
perpendicular, but on the other side of it. 

Let A be the given point, BC the given straight line; let AD be 
the perpendicular from A on BC, 
and AE, AF any two obliques of 
which AF makes the greater angle 
inihAD. 

Produce AD to A', making A'D 
equal to AD, and join A'E, A F. 

Then the triangles ADE, ADE 
are equal in all respects ; and so are 
the triangles ADF, A'DF 

Now (i) in the triangle AEA' the 
two sides AE, EA are greater than AA [i. 20J, that is, twice AE is greater 
^lan twice AD. 

19 — 2 




392 BOOK I [mi, 99 

Therefore A£ is greater than AD. 

(2) Since AE, AE are drawn to £, a point within the triangle AFA\ 

AF, FA are together greater than AE^ EA\ [l 21] 

or twice AFis greater than twice AE. 
Therefore ^^is greater than AE. 

(3) Along DB measure off DG equal to DF^ and join AG. 

The triangles AGD, AFD are then equal in all respects, so that the 
angles GAD^ FAD are equal, and AG is equal to AP. 

Proposition 22. 

Out of three straight lines, which are equal to three given 
straight lines, to construct a triangle : thus it is necessary that 
two of the straight lines taken together in any manner should 
be greater than the remaining one. [i. 20] 

Let the three given straight lines be A, B, C, and of these 
let two taken together in any manner be greater than the 
remaining one, 

namely A, B greater than C, 

A, C greater than B, 
and B, C greater than A ; 

thus it is required to construct a triangle out of straight lines 
equal to A, B, C. 




A- 

B- 
C- 



Let there be set out a straight line DE, terminated at D 
but of infinite length in the direction of E, 
and let DF be made equal to A, FG equal to B, and GH 
equal to C. [i- 3] 

With centre F and distance FD let the circle DKL be 
described ; 

again, with centre G and distance GH let the circle KLH be * 
described ; 
and let KF, KG be joined ; 

I say that the triangle KFG has been constructed out of 
three straight lines equal to A, B, C. 



i I. at] PROPOSITIONS ai, 22 293 



( 

( 



For, since the point /^ is the centre of the circle DKL, 

FD is equal to FK. 
But FD is equal to A ; 

therefore KF is also equal to A. 
Again, since the point G is the centre of the circle LKH^ 

GH is equal to GK. 
But GH is equal to C \ 

therefore KG is also equal to C 
And FG is also equal to B ; 
therefore the three straight lines KF, FG, GK are equal to 
the three straight lines A, B, C. 

Therefore out of the three straight lines KF, FG, GK, 
which are equal to the three given straight lines A, B, C, the 
triangle KFG has been constructed. 

^ Q. E. F. 

3 — 4. This is the first case in the Eltments of a iiapifffjM to a problem in the sense of a 
statement of the conditions or limits of the possibility of a solution. The criterion is of 
coarse supplied by the preceding proposition. 

3. thus it is necessary, lliis is usaally translated (e.g. by Williamson and Simson) 
**£ui it is necessary,'' which is however inaccurate, since the Greek is not M 84 but 8«t d^. 
The words are the same as those used to introduce the tiopw^Lbi in the other sense of the 
"definition" or "particular statement** of a construction to be effected. Hence, as in the 
latter case we say "thus it is required *' (e.g. to bisect the finite straight line AB, i. lo), we 
should here translate *' thus it is necessary.^ 

4. To this enunciation all the mss. and Boethius add, after the iiOfMiiM^ the words 
"because in any triangle two sides taken together in any manner are greater than the 
remaining one.** But ub explanation has the appearance of a gloss, and it is omitted by 
Produs and Campanus. Moreover there is no corresponding addition to the iwpLrikbt 
of VI. «8. 

It was early observed that Euclid assumes, without giving any reason, that 
the circles drawn as described will meet if the condition that any two of the 
straight lines A, B, C are together greater than the third be fulfilled. Proclus 
(p- 33I9 S sqq.) argues the matter by means of reductio ad absurdum, but 
does not exhaust the possible hypotheses inconsistent with the contention. 
He says the circles must do one of three things, (i) cut one another, (2) touch 
one another, (3) stand apart (Sicoravat) from one another. He then considers 
the hypotheses {a) of their touching externally, (p) of their being separated 
from one another by a space. He should have considered also the hypothesis 
{f) of one circle touching the other internally or lying entirely within the 
other without touching. These three hypotheses being successively disproved, 
it follows that the circles must meet (this is the line taken by Camerer and 
Todhunter). 

Simson says: "Some authors blame Euclid because he does not 
demonstrate that the two circles made use of in the construction of this 
problem must cut one another : but this is very plain from the determination 
he has given, namely^ that any two of the straight lines DF, FG, GHmyi%\. 
be greater than the third. For who is so dvdl, though only beginning to 
learn the Elements, as not to perceive that the circle described from the 
centre F, at the distance FD, must meet FH betwixt F and H, because FD 
18 less than FH\ and that, for the like reason, the drcle described from the 



394 BOOK I [i. 33, 33 

centre G at the distance GH must meet DG betwixt D and G ; and that 
these circles must meet one another, because FD and GH are together 
greater than FG:' 

We' have in fact only to satisfy ourselves that one of the drcles, e.g. that 
with centre G^ has at least one point of its circumference outside the other 
circle and also at least one point of its circumference inside the same circle ; 
and this is best shown with reference to the points in which the first circle 
cuts the straight line BE. For ^i) FH^ being equal to the sum of B and C, 
is greater than A^ i.e. than the radius of the circle with centre F^ and therefore 
H is outside that circle. (2) If GM be measured along (7^ equal to GH 
or C, then, since GM is either (a) less or greater than GF, JVwill fidl 
either {a) between G and ^or {b) beyond /^towards Z>; in the first case 
{a) the sum of FM and C is equal to FG and therefore less than the sum 
of A and C, so that FM is less than A or FD; in the second case {i) the 
sum of MF and FG^ i.e. the sum of MF and By is equal to GMix C «m1 
therefore less than the sum of A^ and B^ so that MF is less than A ^ FD\ 
hence in either case M &Us within the circle with centre F. 

It being now proved that the circumference of the circle with centre G 
has at least one point outside, and at least one point inside, the drde with 
centre F^ we have only to invoke the Principle of Continm'ty, as we have to 
do in I. I (cf. the note on that proposition, p. 242, where the necessary 
postulate is stated in the form suggested by Killing). 

That the construction of the proposition gives only two points of 
intersection between the circles, and therefore only two triangles satisfying 
the condition, one on each side of FG^ is dear firom i. 7, which, as bdbre 
pointed out, takes the place, in Book i., of iil 10 proving diat two ciides 
cannot intersect in more points than two. 

Proposition 23. 

On a given straight line and at a point an it to construct a 
rectilineal angle equal to a given rectilineal angle. 

Let AB be the given straight line, A the point on it, and 
the angle DCE the given rectilineal angle; # 

thus it is required to construct on tne given straight line 
AB, and at the point A on it, a rectilineal angle equsu to the 
given rectilineal angle DCE. 





On the straight lines CD, CE respectively let the points 
/?,.£* be taken at random ; 
let DE be joined, 
arid out of three straight lines which are equal to the three 



I. 23] PROPOSITIONS 22, 23 29s 

Straight lines CD, DE, CE let the triangle AFG be con- 
structed in such a way that CD is equal to AF, CE to AG, 
and further DE to FG. ^i. 22] 

Then, since the two sides DC, CE are equal to the two 

^ sides FA, AG respectively, 

' and the base DE is equal to the base FG, 

the angle DCE is equal to the angle FAG. [i. 8] 

\ Therefore on the given straight line AB, and at the point 

I -^ on it, the rectilineal angle FAG has been constructed equal 

f to the given rectilineal angle DCE. 

This problem was, according to Eudemus (see Proclus, p. 333, 5), "rather 
the discovery of Oenopides," from which we must apparently infer, not that 
Oenopides was the first to find any solution of it, but that it was he who dis- 
covered the particular solution given by Euclid. (Cf. Bretschneider, p. 65.) 

The editors do not seem to have noticed the fact that the construction of 
the triangle assumed in this proposition is not exactly the construction given 
in I. 22. We have here to construct a triangle on a certain finite straight line 
AG9& base; in i. 22 we have only to construct a triangle with sides of given 
length without any restriction as to how it is to be placed. Thus in i. 22 we 
set out any line whatever and measure successively three lengths along it 
b^inning from the given extremity, and what we must r^ard as the base is the 
intermecUate length, not the length banning at the given extremity of the 
straight line arbitrarily set out Here the base is a given straight line abutting 
at a given point Thus the construction has to be modified somewhat from 





I 



ITb 



that of the preceding proposition. We must measure AG along AB so that 
AG\& equal to CE (or CD), and C^ along GB equal to DE\ and then we 
must produce BA, in the opposite direction, to ^ so that AF\& equal to CD 
(or CE, ii AG has been made equal to CD). 

Then, by drawing circles (i) with centre A and radius AF, (2) with centre 
G and radius GIf, we determine J^, one of their points of intersection, and we 
prove that the triangle J^AG is equal in all respects to the triangle DCE, and 
then that the angle at A is equal to the angle DCE. 

I think that Proclus must (though he does not say so) have felt the same 
difficulty with regard to the use in i. 23 of the result of i. 22, and that this is 
probably the reason why he gives over again the construction which I have 
given above, with the remark (p. 334, 6) that "you may obtain the construction 
of the triangle in a more instructive manner (StSaoncaXaci^cpov) as follows." 

Proclus objects to the procedure of Apollonius in constructing an angle 
under the same conditions, and certainly, if he quotes Apollonius correctly, &e 
latter's exposition must have been somewhat slipshod. 





S96 BOOK I [i. S3» 94 

"He takes an angle CDE at random," says Produs (p. 335, 19 sqq.X ''and 
a straight line AB^ and with centre D and distance 
CD describes the circumference C£, and in die same 
way with centre A and distance AB the circumference 
FB. Then, cutting off FB equal to CE, he joins AF. 
And he declares that the aisles A^ D standii^ on 
equal circumferences are equaL" _ 

In the first place, as Proclus remaiks, it should be E 

premised that AB is equal to CD in order that the ^^^ 

circles may be equal; and the use of Book in. for 
such an elementary construction is objectionable. 
The omission to state that AB must be taken equal 
to CD was no doubt a slip, if it occurred. And, as 
r^ards the equal angles '' standing on equal drcum- . ^ 

.ferences," it would seem possible mat Apollonius said 

this in explanation^ for the siJce of brevity, rather than by way of proof. It 
seems to me probable that his construction was only given from the point of 
view olfrachcal^ not theoretical, geometry. It really comes to the same thing 
as Euclid's except that DC is taken equal to DE. For cutting off the arc BF 
equal to the arc CE can only be meant in the sense of measuring the M^n/ 
C£, say, with a pair of compasses, and dien drawing a circle with centre B 
and radius eqiial to the chord CE. Apollonius' direction was therefore 
probably intended as a practical short cut, avoiding the actual drawing of the 
chords CE^ BF^ which, as well as a proof of the ec^uali^ in all respects of the 
triangles CDE^ BAF^ would be required to estabhsh tluonikaUy the correct- 
ness of the construction. 



Proposition 24. 

If two triangles have the two sides equal to two sides 
respectively^ but have the one of the angles contained by the equal 
straight lines greater than the other, they will also have the 
bc^e greater than the base. 

S Let ABC, DEF be two triangles having the two sides 

AB^ -^C equal to the two sides DE, Z?/^ respectively, namely 

AB to DE, and -^C to DF, and let the angle at -^ be greater 

than the angle at D ; 

I say that the base BC is also greater than the base EF. 
10 For, since the angle BAC 

is greater than the angle EDF, 

let there be constructed, on the 

straight line DE, and at the 

point D on it, the angle EDG 
IS equal to the angle BAC; [i. 23] 

let DG be made equal to either 

of the two straight lines AC, 

DF, and let EG, FG be joined. 





I 

I 

' 1.24] PROPOSITIONS 23,24 297 

i Then, since AB is equal to DE, and AC to DG, 

j ao the two sides BA, AC are equal to the two sides ED, DG, 

respectively ; 
. and the angle BAC is equal to the angle EDG ; 

therefore the base BC is equal to the base EG. [1. 4] 
V Again, since DF is equal to DG, 

* 25 the angle DGF is also equal to the angle DFG ; [i. 5] 

therefore the angle DFG is greater than the angle EGF. 
It Therefore the angle EFG is much greater than the angle 

* EGF. 

And, since EFG is a triangle having the angle EFG 
30 greater than the angle EGF, 

and the greater angle is subtended by the greater side, 

[I. 19] 
the side EG is also greater than EF. 
But EG is equal to BC. 

Therefore BC is also greater than EF. 
f 35 Therefore etc. 

Q. E. D. 

10. I have naturally left out the well-known words added by Simson in 
order to avoid the necessity of considering three cases : " Of the two sides 
DE, DF let DE be the side which is not greater than the other." I doubt 
whether Euclid could have been induced to insert the words himself, even if 
it had been represented to him that their omission meant leaving two possible 
cases out of consideration. His habit and that of the great Greek geometers 
was, not to set out all possible cases, but to give as a rule one case, generally 
the most difficult, as here, and to leave the others to the reader to work out for 
himself. We have already seen one instance in i. 7. 

Proclus of course gives the other 
two cases which arise if we do not 
first provide that DE is not greater 
than DF. 

(i) In the first case G may fall 
on EF produced, and it is then 
obvious that EO is greater than EF, 

(2) In the second case EG may 
fall below EF. 

If so, by 1. 21, DF, FE are 
together less than DG, GE. 

But DF is equal to DG \ there- 
fore EF is less than EG, i.e. than 
BC. 

These two cases are therefore 
decidedly simpler than the case taken 
by Euclid as typical, and could well be left to the ingenuity of the learner. 

If however after all we prefer to insert Simson's words and avoid the latter 




2gB 



BOOK I 



[1.24 



two cases, the proof is not complete unless we show that, with his assumptioo, 
/^must, in the figure of the proposition, &11 Mn» EG. 

De Morgan would make the following proposition precede: Every siraigki 
line drawn from the vertex of a trmngk to the base is less than the greaier of ike 
two sides, or than either if they are equal, and he would prove it by means of 
the proposition relating to perpendicular and obliques mven above, p. 391. 

But it is easy to prove directly that F &lls below EG, if 
DE is not greater than DG^ by the method employed by 
Pfleiderer, Lardner, and Todhunter. 

Let DF, produced if necessary, meet EG in H. 

Then the angle DHG is greater than the angle DEG\ 

[I. 16] 

and the angle DEG is not less than the angle DGE : 

[.. 18] 

therefore the angle DHG is greater than the angle DGH. 
Hence DH\% less than DG^ [i. 19] 

and therefore DH is less than DF. 

Alternative proof. 

Lastly, the modem alternative proof is worth giving. 

A D A 





Let Z>^ bisect die angle FDG (after the triangle DEG has been made 
equal in all respects to the triangle ABC^ as in the proposition), and let DH 
meet EG in H Join HF. 

Then, in the triangles FDH, GDH, 

the two sides FD, DHzx^ equal to the two sides GD, DH, 
and the included angles FDH, GDHzxt, eqiial ; 
therefore the base HF is equal to the base HG. 
Accordingly EG is equal to the sum of EH, HF\ 

and EH, HF are together greater than EF\ [i. 20] 

therefore EG, or BC, is greater than EF. 
Proclus (p. 339, 1 1 sqq.) answers by anticipation the possible question that 
might occur to any one on this propc^ition, viz. why does Euclid not compare 
the areas of the triangles as he does in i. 4 ? He observes that inequali^ of 
the areas does not follow from the inequality of the angles contained by the 
equal sides, and that Euclid leaves out all reference to the question boUi for 
this reason and because the areas cannot be compared without the help of the 
theory of parallels. *' But if^" says Proclus, *' we must anticipate what is to 
come and make our comparison of the areas at once, we assert that (i) if 
the an^ Af D — stiptposing that our argument proceeds with reference to the 
figure in the proposition — are {together) equeU to two right angles, the triangles 



I. 24, 25] PROPOSITIONS 24, 25 299 

are proved equals (2) if greater than two right angles^ that triangie which has 
the greater angle is lesSy and (3) if they are less, greater" Proclus then gives 
the proof, but without any reference to the source from which he quoted 
the proposition. Now an-Nairiz! adds a similar proposition to i. 38, but 
definitely attributes it to Heron. I shall accordingly give it in the place 
where Heron put it 



Proposition 25. 

// two triangles have the two sides equal to two sides 
respectively, but have the base greater than the base, thiy will 
also have the one of the angles contained by the equal straight 
lines greater than the other. 

Let ABC, DEF be two triangles having the two sides 
AB^ AC equal to the two sides DE^ Z?/^ respectively, namely 
AB to DE, and -^C to DF\ and let the base BC be greater 
than the base EF\ 

I say that the angle BAC is also greater than the angle 
EDF. 





For, if not, it is either equal to it or less. 
• Now the angle BAC is not equal to the angle EDF\ 

I for then the base BC would also have been equal to the base 
, EF, [1.4] 

but it is not ; 

therefore the angle BAC is not equal to the angle EDF. 

Neither again is the angle BAC less than the angle EDF; 

for then the base BC would also have been less than the base 

EF, ^ [1.24] 

but it is not ; 
therefore the angle BAC is not less than the angle EDF. 
But it was proved that it is not equal either ; 

therefore the angle BAC is greater than the angle EDF. 
Therefore etc. 

Q. E. D. 



300 BOOK I [1.95 

De Morgan points out that ibis jvoposition (as also i. 8) is a purdy iogieai 
consequence of i. 4 and i. 34 in the same way as i. 19 and l 6 are purdy 
icgua/ consec^uences of i. 18 and 1. 5. If n, ^, c denote the sidesi A^ B, Cytid , 
angles opposite to them in a triangle ABQ and a^^ V^ /, A\ B, Ciht sides* 
and opposite angles respectivdy in a triangle A'BC^ i. 4 and i. 24 tdl us 
that, b^ c being respectively equal to V^ i^ 

(i) if ^ is equal to A\ then a is equal to a\ 

(2) if ^ is less than A\ then a is less than a', 

(3) if ^ is greater than A\ then a is greater than a' ; 
and it follows iogually that, 

(i) if a is equal to a', the angle A is equal to the angle A\ \u 8] 

(3) if a is less than a', ^ is less than A\ \ . . 

(3) if a is greater than «*, -^ is greater than A\ ] L'' *5j 

Two alternative proofs of this theorem are given by Produs (pp. 345 — 7), 
and they are both interesting. Moreover both are Hred. 

I. Proof by Menelaus of Alexandria. 

Let ABC, jDEFbe two triangles having the two sides BA, AC equal to 
the two sides EjD, DF, but the base ^C greater than the base EF. 





Then shall the angle at ^ be greater than the angle at Z>. 
From BC cut off BG equal to EF. At B, on the straight line BC, make 
the angle GBH {on the side of BG remote from A) equal to the angle FED. 
Midce BH equal to DE\ join HG, and produce it to meet AC in K. 
Join AH. 

Then, since the two sides GB, BH are equal to the two sides FE^ ED 
respectively, 

and the angles contained by them are equal, 
HG is equal to DFox AC^ 



\ 



and the angle BHG is equal to the angle EDF. ^ 

Now HK is greater than HG 01 AC, \ 

and a fortiori greater than AK\ 

therefore the angle KAH\& greater than the angle KHA. 

And« since AB is equal to BH, 

the angle BAH\% equal to the angle BHA. 

Therefore, by addition, 

the whole angle BA C is greater than the whole angle BHG, 
that is, greater than the angle EDF. 



1. 25, a6] PROPOSITIONS 25, 26 301 

II. Heron's proof. 

Let the triangles be given as before. 

Since BC is greater than EF^ produce EF to (r so that EG is equal to 
BC. 

Produce ED to If so that DIf is equal to I?F The circle with centre 
D and radius Z>^will then pass through If. Let it be described^ as FXIf. 




Now, since B:,!^ AC axe together greater than BQ 

and BA, AC Bie equal to ED^ />^ respectively, 

while BC is eqiial to EG^ 

EH is greater than EG. 

Therefore the circle with centre j^and radius EG will cut Elf^ and 

therefore will cut the circle already drawn. Let it cut that circle in JT, and 

join DK, KE. 

^ Then, since Z> is the centre of the circle FKH^ 

iDK is eqiial to DF or AC. 
Similarly, since E is the centre of the circle KG^ 
EK\& equal to EG or BC^ 
And DE is equal to AB. 

Therefore the two sides BA^ AC 9st eqiial to the two sides ED^ DK 
respectively ; 

and the base BC is equal to the base EK\ 
therefore the angle BAC is equal to the angle EDK. 
Therefore the angle BAC is greater than the angle EDF. 



Proposition 26. 

I If two triangles have the two angles equal to two angles 
I respectively y and one side equal to one side^ namely, either the 
side adjoining the equal angles, or thai subtending one of the 
equal angles, they will also have the remaining sides equal to 
5 the remaining sides and the remaining angle to the remaining 
angle. 



302 BOOK I [l s6 

Let ABC, DEF be two triangles having the two angles 
ABC, BCA equal to the two angles DEF, EFD respectivelv, 
namely the angle ABC to the angle DEF, and the ansle 

^^BCA to the angle EFD; and let them also have one side 
equal to one side, first that adjoining the equal angles, namely 
BC to EF\ 

I say that they will also have the remaininc; sides equal 
to the remaining sides respectively, namely aS to DE and 

15-^C to DF, and the remaining angle to the remaining angle, 
namely the angle BAC to the angle EDF. 





B 

For, if AB is unequal to DE, one of them is greater. 
Let AB be greater, and let BG be made equal to DE ; 
and let GC be joined. 
ao Then, since BG is equal to DE, and BC to EF, 
the two sides GB, BC are equal to the two sides DE, EF 
respectively; 
and the angle GBC is equal to the angle DEF; 

therefore the base GC is equal to the base DF, 
25 and the triangle GBC is equal to the triangle DEF, 
and the remaining angles will be equal to the remaining angles, 
namely those which the equal sides subtend ; [i. 4] 

therefore the angle GCB is equal to the angle DFE. 
But the angle DFE is by hypothesis equal to the angle BCA ; 
30 therefore the angle BCG is equal to the angle BCA, 

the less to the greater : which is impossible. 
Therefore AB is not unequal to DE, 
and is therefore equal to it. 
But BC is also equal to EF-, 
35 therefore the two sides AB, BC are equal to the two 

sides DE, EF respectively, 
and the angle ABC is equal to the angle DEF\ 

therefore the base ACv& equal to the base /?/% 
and the remaining angle BAC is equal to the remaining 
40 angle EDF. [l 4] 



T 



26] PROPOSITION 36 303 

Again, let sides subtending equal angles be equal, as AB 
, to DE\ 

' I say again that the remaining sides will be equal to the 

remaining sides, namely AC to DF and BC to EF, and 

^45 further the remaining angle BAC is equal to the remaining 

angle EDF. 
1 For, if BC is unequal to EF^ one of them is greater. 

Let BC be greater, if possible, and let BH be made equal 
to EF\ let Am be joined. 
50 Then, since Bm is equal to EF, and AB to DE, 
the two sides AB, BH are equal to the two sides DE, EF 
respectively, and they contain equal angles ; 

therefore the base AH is equal to the base DF, 
and the triangle ABH is equal to the triangle DEF, 
55 and the remaining angles will be equal to the remaining angles, 
namely those which the equal sides subtend ; [i. 4] 

therefore the angle BHA is equal to the angle EFD. 
But the angle EFD is equal to the angle BCA ; 
therefore, in the triangle AHC, the exterior angle BHA is 
60 equal to the interior and opposite angle BCA : 

which is impossible. [i. 16] 

Therefore BC is not unequal to EF, 

and is therefore equal to it. 
But AB is also equal to DE ; 
65 therefore the two sides AB, BC are equal to the two sides 
DE, EF respectively, and they contain equal angles ; 
therefore the base -^C is equal to the base DF, 
the'triangle ABC equal to the triangle DEF, 
and the remaining angle BAC equal to the remaining angle 
10 EDF. [1.4I 

Therefore etc. 

rt Q. E. D. 



f 



1 — 3. the side adjoining the equal angles, vXevpdr rV vp^ Ta7f fo-cuf 'pmUm. 

99. is by hjrpothesis equal. Imhtntrnx C^, aocording to the elegant Greek idiom. 
iwUutiM is used for the passive of (nrcrrlBiiiu^ as iretMoi is used for the passive of rlBiifu, and 
io with the other oompoonds. Cf. vpoffKOffieu, ** to be added.'* 

The altemative method of proving this proposition, viz. by applying one 
triangle to the other, was very early discovered, at least so far as r^utls the 
case where the equal sides are adjacent to the equal angles in each. ^-NairizI 
gives it for this case, observing that the proof is one which he had found, but 
of which he did not know the author. 



ii 




304 BOOK I [I. ,6 

Proclus has the following interesting note (p. 359, 13 — 18): "Eudemiu 
in his geometrical history refers this theorem to Thales. For he says that» in 
the method by which they say that Thales proved the distance of ships in the 
sea, it was necessary to make use of this theorem." As, wiibrtunatdy, this 
information is not sufficient of itself to enable us to determine how Thales 
solved this problem, there is considerable room for conjecture as to his 
method. 

The suggestions of Bretschneider and Cantor agree in the assumption 
that the necessary observations were probably made from the top oi some 
tower or structure of known height, and that a right-angled triangle was used in 
which the tower was the perpendicular, and the line connecting the bottom of 
the tower and the ship was the base, as in the annexed figure, where AB is the 
tower and C the ship. Bretschneider {Die Geometrie und die Geometer par 
EukUides^ % 30) says that it was only necessary for 
the observer to observe the angle CAB^ and then 
the triangle would be completely determined by 
means of this angle and the known length AB. 
As Bretschneider says that the result would be 
obtained '' in a moment " by this method, it is not 
clear in what sense he supposes Thales to have 
"observed" the angle BAC. Cantor is more 
definite (GescK d. Math, i„ p. 145), for he says that 
the problem was nearly related to that of finding the 
Seqt from given sides. By the Seqt in the Papyrus Rhind is meant, according 
to the conjecture of Cantor and Eisenlohr, a number representing the ratio to 
one another of the lengths of certain lines in pyramids or obelisks ; sometimes 
it is practically equivalent to the cosine of the angle made by the sloping edge 
of a pyramid and the semi-diagonal of the base, sometimes to the tangent of 
the angle made by the perpendicular from the vertex of the pyramid on one 
side of the base and the line connecting the foot of that perpendicular and the 
centre of the base. The calculation of the Seqt thus implying a sort of theory 
of similarity, or even of trigonometry, the suggestion of Cantor is apparenUy 
that the Seqt in this case would be found from a small right-angled triangle 
ADE having a common angle A with ABC as shown in the figure, and that 
the ascertained value of the Seqt with the length AB would determine BC. 
This amounts to the use of the property of similar triangles; and 
Bretschneider's suggestion must apparenUy come to the same thing, since, 
even if Thales measured the angle in our sense (e.g. by its ratio to a right 
angle), he would, in the absence of something corresponding to a table of 
trigonometrical ratios, have gained nothing and would have had to work out 
the proportions all the same. 

Max C. P. Schmidt also {Kulturhistorische Beitrdge %ur Kenntnis des 
griechischen und rihnischen Altertums^ 1906, p. 32) similarly supposes Thales to 
have had a right angle made of wood or bronze with the legs graduated, to 
have placed it in the position ADE (A being the position of his eye), and 
then to have read ofi* the lengths AD, DE respectively, and worked out the 
length of BC by rule of three 

How then does the supposed use of similar triangles and their property 
square with Eudemus' remark about i. 26? As it stands, it asserts the 
equality of ttvo triangles which have two angles and one side respectively 
equal, and the theorem can only be brought into relation with the above 
explanations by taking it as asserting that, if two angles and one side of one 
triangle are given, the triangle is completely determined. But, if Thales 




I. 26] PROPOSITION 26 305 

/ practically Msed froportionsy as supposed, i. 26 is surely not at all the theorem 
which this proc^ure would naturally suggest as underlying it and being 

( *' necessarily used"; the use of proportions or of similar but not equal 

^ triangles would surely have taken attention altogether away from i. 26 and 
fixed it on vi. 4. 

) For this reason I think Tannery is on the right road when he tries to find 

a solution using i. 26 as it stands, and withal as primitive as any recorded 
solution of such a problem. His suggestion (La Giomitrie gncque^ pp. 90—1) 

* is based on the fluminis varaiio of the Roman agrimensor Marcus Junius 
Nipsus and is as follows. 

iTo find the distance from a point ^ to an inaccessible point B, From A 
measure along a straight line at right angles to AB a 
length i^C and bisect it at Z>. From C draw CE at right 
angles to CA on the side of it remote from B^ and let E 
be the point on it which is in a straight line with B and D, 
Then, by 1. 26, CE is obviously equal to AB, 
As r^ards the equality of angles, it is to be observed 
that those at D are equal beotuse they are vertically 
opposite, and, curiously enough, Thales is expressly 
^ credited with the discovery of the equality of such angles. 
The only objection which I can see to Tannery's 
' solution is that it would require, in the case of the ship, a 
certain extent of fi-ee and level ground for the construction 
and measurements. 

I suggest therefore that the following may have been 
Thales' method. Assuming that he was on the top of a 
tower, he had only to use a rough instrument made of a straight stick and a 
cross-piece fastened to it so as to be capable of turning about the fastening 
(say a nail) so that it could form any angle with the stick and would remain 
where it was put. Then the natural thing would be to fix the stick upright 
(by means of a plumb-line) and direct the cross-piece towards the ship. 
Next, leaving the cross-piece at the angle so found, the stick could be turned 
round, still remaining vertical, until the cross-piece pointed to some visible 
object on the shore, when the object could be mentally noted and the distance 
^ from the bottom of the tower to it could be subsequently measured. This 
! would, by i. 26, give the distance from the bottom of the tower to the ship. 
i This solution has the advantage of corresponding better to the simpler and 
more probable version of Thales' method of measuring the height of the 
pyramids; Diogenes Laertius says namely (i. 27, p. 6, ed. Cobet) on the 
authority of Hieronymus of Rhodes (b.c 300 — 260), that he waited for this 
purpose until the moment when our shadows are of the same length as ourselves. 

Recapitulation of congruence theorems. 

Proclus, like other commentators, gives at this point (p. 347, 20 sqq.) a 
summary of the cases in which the equality of two triangles m all respects can 
be established. We may, he says, seek the conditions of such equality by 

(successively considering as hypotheses the eqiiality (i) of sides alone, (2) of 
angles alone, (3) of sides and angles combined. Taking (i) first, we can only 
establish the equality of the triangles in all respects if all three sides are 
respectively equal; we cannot establish the equality of the triangles by any 
hypothesis of class (2), not even the hypothesis that all the three angles are 
respectively eqiial ; among the hypotheses of class (3), the equality of one 

H. E. 20 



3o6 BOOK I [i.a6 

side and one angle in each triangle is not enough, the equality (a) ci one side 
and all three angles is more than enough, as is also the equali^ {i) of two 
sides and two or three angles, and {c) of three sides and one or two an^^ 

The only hypotheses therefore to be eramined firom this point of view aie 
the equality of 

(a) three sides [EucL l 8]. 

03) two sides and one angle ^i. 4 proves one case of this, where the an^ 
is that contained by the sides which are by hypothesis equal]. 

(y) one side and two angles [i. 36 covers all cases]. 

It is curious that Proclus makes no allusion to what we call the ambiguams 
case^ that case namely of 09) in which it is an angle opposite to one ^tbe 
two specified sides in one triangle which is ecjual to the angle opposite to the 
equal side in the other triangle. Camerer indeed attributes to Proclus the 
observation that in this case the equality of the triangles caimot, unless scmie 
other condition is added, be asserted generally; but it would appear that 
Camerer was probably misled by a figure (Proclus, p. 351) which looks like a 
figure of the ambiguous case but is really only used to show that in l 26 the 
equal sides must be camspandrng sides, i.e., they must be either adjacent to the 
equal angles in each triangle, or opposite to corresponding equal angles, and 
that, e.g., one of the equd sides must not be adjacent to the two angles in 
one triangle, while the side equal to it in the other triangle is opposite to one 
of the two corresponding angles in that triangle. 

The ambiguous case. 

If two friangies have two sides equal io two sides respecHveiy^ and if ike 
angles opposite to one pair of equal sides he also equals then will ike angles 
opposite the other pair of equal sides be either equal or suppUmmtary ; emd^ in 
the former case^ the triangles will be equal in all respects. 

Let ABC, DEFht two triangles such that AB is equal to DE, and AC 
to DE, while the angle ABC is equal to the angle DEF\ 
it is required to prove that the angles ACB, DFE are either equal or 
supplementary. 



^ A D 




Now (i), if the angle BAC be equal to the angle EDF, it follows, since 
the two sides AB^ AC Site equal to the two sides !?£, 27^ respectively, that . 
the triangles ABC, DEFdj^ equal in all respects, [i. 4] 

and the angles ACB, DFE are equal. 

(2) If the angles BAC, EDF be not equal, make the angle EDG (on 
the same side of ED as the angle EDF) equal to the angle BAC 

Let EF, produced if necessary, meet DG in G. 

Then, in the triangles ABC, DEG, 
the two angles BAC, ABC are equal to the two angles EDG, DEG 
respectively, 
and die side AB is equal to the side DE ; 



J 



I. 26, 27] PROPOSITIONS 26, 27 307 

therefore the triangles ABC^ DEG are equal in all respects, [i. 26] 
so that the side ^C is equal to the side DG^ 
and the angle ACB is equal to the angle DGE. 
Again, since AC is eqiial to DF9& well as to DG^ 
DF\& equal to DG, 
and therefore the angles DFG^ DGF^xt, equal. 

But the angle DFE is supplementary to the angle DFG\ and the angle 
DGF^N^s proved equal to the angle ACB; 

therefore the angle I?F£ is supplementary to the angle ACB. 
If it is desired to avoid the ambiguity and secure that the triangles may 
be congruent, we can introduce the necessary conditions into the enunciation, 
on the analogy of Eucl. vi. 7. 

Jf (tvo triangles have two sides of the one equal to two sides of the other 
) respectively^ and the angles opposite to a pair of equal sides equals then, if the 
i angles opposite to the other pair of equal sides are both acute^ or hoth obtuse^ or if 
one of them is a right angle^ the two triangles are equal in all respects. 

(The proof of the three cases (by reductio cui absurdum) was given by 
Todhunter. 



Proposition 27. 

If a straight line falling on two straight lines make the 
alternate angles equal to one another, the straight lines mill be 
parallel to one another. 

% For let the straight line EF falling on the two straight 

5 lines AB, CD make the alternate angles AEF, EFD equal 
^ to one another ; 

I say that AB is parallel to CD. 
For, if not, AB, CD when pro- 
duced will meet either in the direction 
\o of B, D or towardis A, C. 

Let them be produced and meet, 
in the direction of B, Z>, at G. 
* Then, in the triangle GEF, 

the exterior angle A EF is equal to the interior and opposite 
15 angle EFG : 
which is impossible. [i. 16] 

Therefore AB, CD when produced will not meet in the 
direction of B, D. 

Similarly it can be proved that neither will they meet 
o towards A, C 

20 — 2 



i 




3o8 BOOK I [i. «7 

But straight lines which do not meet in either direction 
are parallel ; [Dcf. 23] 

therefore AB is parallel to CD. 
Therefore etc. 

Q. E. D. 

I. falling on two straight lines, 4,% Mo MtU% kiarifmntti^ the phnse being the flune 
as that used in Post. 5, meaning a trtmsversml* 

3. the alternate angles, oi ^miXXAI Twrku. Produt fp. ^57, 9) expfauns that Eadid 
uses the word aitemate (or, more exactly, aiiimatefy^ ^roXXd^ m two connexions, (i) of a 
certain transformation of a proportion, as in Book V. and the arithmetical Books, (s) as here, 
of certain of the angles formed by parallels with a straight line crossing them. AlUnmU 
angles are, according to Euclid as interpreted by Prodns, those which are not on the same 
side of the transveruil, and are not adjacent, but are separated by the transversal, both being 
within the parallels but one ** above ** and the other ** below." The meaning is natoru 
enough if we imagine the four internal angles to be taken in cyclic order and aH^nmU angkf 
to be any two of them not successive but separated by one angle of the four.' 

9. in the direction of B, D or towards A, C, literally ** towards the farts BtD ot 
towards A, C," M rii B, A fidpti 41 M rii A, T. 

With this proposition b^;ins the second section of the first BodL Up 
to this point Eudid has dealt mainly with triangles, their constmcdcm 
and their properties in the sense of the relation of their parts, the tides and 
angles, to one another, and the comparison of different triangles in respect oi 
their parts, and of their area in the particular cases where they are congruent 

The second section leads up to the third, in which we pass to rdations 
between the areas of triangles, parallelograms and squares, the special feature 
being a new conception of equality of areas, equality not dependent on 
congruence. This whole subject requires the use of parallels. Consequently 
the second section beginning at i. 27 establishes the theory of parallels, 
introduces the cognate matter of the equality of the sum of Uie angles of a 
triangle to two right angles (i. 32), and ends with two propositions forming the ^ 
transition to the third section, namely i. 33, 34, which introduce the parallelo- y 
gram for the first time. 

Aristotle on parallels. ] 

We have already seen reason to believe that Euclid's personal contribution ' 
to the subject was nothing less than the formulation of the famous Postulate 
5 (see the notes on that Postulate and on Def. 23), since Aristotle indicates 
that the then current theory of parallels contained a petiiio prindpii^ and . ! 
presumably therefore it was EucUd who saw the defect and proposed the ^ 
remedy. 

But it is clear that the propositions i. 27, 28 were contained in earlier 
text-books. They were familiar to Aristotle, as we may judge from two j 
interesting passages. I 

(i) In Anal. Post i. 5 he is explaining that a scientific demonstration 
must not only prove a fact of every individual of a class (#car& wavro^) but ' 
must prove it primarily and generally true (irp«;^ov #ca^<^Xov) of the whole of 
the class as one ; it will not do to prove it first of one part, then of another . 
part, and so on, until the class is exhausted. He illustrates this (74 a 13 — 16) I 
by a reference to parallels : '' If then one were to show that right (angles) do 
not meet, the proof of this might be thought to depend on the fact diat this 
is true of all (pairs of actual) right angles. But this is not so, inasmuch as 
the result does not follow because (the two angles are) eqtial (to two right 



r 



1. 27, 28] PROPOSITIONS 27. 28 309 

angles^ in the particular way [i.e. because each is a right angle], but by virtue 
i of their being equal (to two right angles) in any way whatever [i.e. because 
I the sum only needs to be equal to two right angles, and the angles themselves 

may vary as much as we please subject to this]." 
' (2) The second passage has already been quoted in the note on Def. 23 : 

' *' there is nothing surprising in different hypotheses leading to the same false 
^ conclusion ; e.g. the conclusion that parallels meet might equally be drawn 
I from either of the assumptions (a) that the interior (angle) is greater than the 
J exterior or {b) that the sum of the angles of a triangle is greater than two 
I right angles" (Anal. Prior, 11. 17, 66 a 11 — 15). 

\ I do not quite concur in the interpretation which Heiberg places upon 

these passages (Mathematisches tu AristoteleSy pp. 18 — 19). He says, first, 
that the allusion to the "interior angle" being "greater than the exterior" in 
I the second passage shows that the reference in the first passage must be to 
Eucl. I. 28 and not to i. 27, and he therefore takes the words ^i wSl Strot in 
the first passage (which I have translated " because the two angles are equal 
to two right angles in the particular way ") as meaning " because the angles, 
viz. the exterior and the interior^ are equal in the particular way." He also 
takes oi ^p^ai ov (rvfixurrovcri (which I have translated " right angles do not 
meet," an expression quite in Aristotle's manner) to mean "perpendicular 
straight lines do not meet " ; this is very awkward, especially as he is obliged 
to supply angles with uroi in the next sentence. 

But I think that the first passage certainly refers to i. 28, although I do 
not think that the alternative {a) in the second passage suggests it. This 
alternative may, I think, equally with the alternative {b) refer to i. 27. That 
proposition is proved by rcductio ad absurdum based on the fact that, if the 
straight lines do meet, they must form a triangle^ in which case the exterior 
angle must be greater than the interior (while according to the hypothesis 
these angles are equal). It is true that Aristotle speaks of the hypothesis 
that the interior anele is greater than the exterior ; but after all Aristotle had 
only to state some incorrect hypothesis. It is of course only in connexion 
with straight lines meetings as the hypothesis in 1. 27 makes them, that the 
alternative {b) about the sum of the angles of a triangle could come in, and 
alternative {a) implies alternative (b). 

It seems clear then from AristoUe that i. 27, 28 at least are pre-Euclidean, 
^d that it was only in i. 29 that Euclid made a change by using his Postulate. 
De Morgan observes that i. 27 is a logical equivalent of i. 16. Thus, if A 
means "straight lines forming a triangle with a transversal," B "straight lines 
making angles with a transversal on the same side which are together less 
than two right angles," we have 

All ^ is ^, 
and it follows logically that 

All not-^ is not-^. 



Proposition 28. 

If a straight line falling on two straight lines make the 
exterior angle equal to the interior and opposite angle on the 
same side, or the interior angles on the same side equal to two 
right angles, Jhe straight lines will be parallel to one another. 





3IO BOOK I [i. 38 

For let the straight line EF falling on the two straight 
lines AB, CD make the exterior angle EGB equal to the 
interior and opposite angle GHD, or the interior angles on 
the same side, namely BGH, GHD, equal to two right angles; : 

I say that AB is parallel to CD. 

For, since the angle EGB is 
equal to the angle GHD, 
while the angle EGB is equal to the 
angle AGH, [i. 15] 

the angle AGH is also equal to the 
angle GHD\ 
and they are alternate ; 

therefore AB is papdlel to CD. [i. 27] 

Again, since the angles BG^.JCrlip^zxt^ equal to two 
right angles, and the angles AGH, BuH are also equal to 
two right angles, [i. 13] 

the angles AGH, BGH are equal to the angles BGH, GHD. 

Let the angle BGH be subtracted from each ; 
therefore the remaining angle AGH is equal to the remaining 
angle GHD\ 
and they are alternate ; 

therefore AB is parallel to CD. [i. 27] 

Therefore etc. j 

Q. E. D. 

One criterion of parallelism, the equality of alternate angles, is given in 
I. 37 ; here we have two more, each of which is easily reducible, and is actually 
reduced, to the other. 

Produs observes (pp. 358 — 9) that Euclid could have stated six criteria as 
well as three, by using, m addition, other pairs of angles 
in the figure (not adjacent) of which it could be predi- 
cated that the two angles are equal or that their sum 
is ec^ual to two right angles. A natural division is to 
consider, first the pairs which are on the same side of 
the transversal, and secondly the pairs which are on 
different sides of it 

Taking (i) the possible pairs on the satne side, we 
may have a pair consisting of 

(a) two internal angles, viz. the pairs (BGH, 
GHD) and (AGH, GHC) ; 

(i) two external angles, viz. the pairs (EGB, DHF) and {EGA, CHF)\ 

(i) one external and one internal angle, viz. the pairs lEGB^ GHD\ 
(FHD, HGS), {EGA, GHC) and {FHC, HGA). 



I. 28, 29] PROPOSITIONS 28, 29 311 

And (2) the possible pairs on different sides of the transversal may consist 
respectively of 

(fl) two internal angles, viz. the pairs {AGH, GHD) and (CZTG, HGE)\ 
\b) two external angles, viz. the pairs {AGE, DHF) and {EGB, CHF)\ 
(f) one external and one internal, viz. the pairs {AGE, GHD\ lEGB, 
GHC\ {FHC, HGB) and {FHD, HGA). 

The angles are equal in the pairs (i) (r), (2) (a) and (2) {b\ and the sum 
is equal to two right angles in the case of the pairs (i) {a\ (i) {p) and (2) (r). 
For his criteria Euclid selects the cases (2) (a) [i. 27J and (i) {c\ (i) (a) [i. 28], 
leaving out the other three, which are of course equivalent but are not quite 
so easily expressed 

From Proclus' note on i. 28 (p. 361) we learn that one Aigeias (? Aineias) 
of Hierapolis wrote an epitome or abridgment of the Elements, This seems 
to be the only mention of this editor and his work; and they are only 
mentioned as having combined Eucl. i. 27, 28 into one proposition. To do 
this, or to make the three hypotheses the subject of three separate theorems, 
would, Proclus thinks, have been more natural than to deal with them, as 
Euclid does, in two propositions. Proclus has no suggestion for explaining 
Euclid's arrangement unless the ground were that i. 27 deals with angles on 
different sides, i. 28 with angles on one and the same side, of the transversal. 
But may not the reason have been one of convenience, namely that the 
criterion of i. 27 is that actually used to prove parallelism, and is moreover 
the basis of the construction of parallels in i. 31, while i. 28 only reduces the 
other two hypotheses to that of i. 27, so that precision of reference, as well as 
clearness of exposition, is better secured by the arrangement adopted ? 

Proposition 29. 

A straight line falling on parallel straight lines makes 
\ the alternate angles equal to one another, the exterior angle 
equal to the interior and opposite angle, and the interior angles 
on the same side equal to two right angles. 
5 For let the straight line EF fall on the parallel straight 
- lines AB, CD ; 

I say that it makes the alternate angles AGH, GHD 
equal, the ekterior angle EGB equal to the interior and 
opposite angle GHD, and the interior angles on the same 
10 side, namely BGH, GHD, equal to two right angles. 
For, if the angle AGH is unequal 
to the angle GHD, one of them is 
greater. 

Let the angle AGH be greater. 
15 Let the angle BGH be added to 
each ; 

therdfore the angles AGH, BGH 
are greater than the angles BGH, 
GHD. 



1' 




31 a BOOK I [h 29 

«> But the angles AGIf, BGH are equal to two right angles; 

therefore the angles BGH, GHD are less than two 
right angles. 

But straight lines produced indefinitely from angles less 

than two right angles meet ; [Post. 5] 

25 therefore AB, CD, if produced indefinitely, will meet ; 

but they do not meet, because they are by hypothesis parallel. 

Therefore the angle AGH is not unequal to the angle 

GHD, 

and is therefore equal to it 
JO Again, the angle AGH is equal to the angle EGB ; [i. 15] 
therefore the angle EGB is also equal to the angle 
GHD. [C. N. i] 

Let the angle BGH be added to each ; 

therefore the angles EGB, BGH are equal to the 

js angles BGH. GHD. [C. N. a] 

But the angles EGB, BGH are equal to two right angles; 

P-I3] 
therefore the angles BGH, GHD are also equal to 
two right angles. 

Therefore etc. q. e. d. 

13. straight lines produced indefinitely from angles less than two right angles, 
al M dr* IkaffoiufWf if h^ 6p$C» iKfioKKhfUPot, e/f dw€ipow cvftwlwrouavt a variation from the 
more explicit language of Postulate 5. A £ood deal is left to be understood, namely that the 
straight lines begin from pointo at which they meet a thmsversal, and make with it internal 
angles on the same side the sum of which is equal to two right angles. 

a6. because they are by hjrpothesis parallel, literally ** because they are supposed 
parallel," Ml t6 ra^M^i^Xovf uMls ihrmrfSytfoi. 

Proof by " Plasrfair's " axiom. 

If, instead of Postulate 5, it is prefen-ed to use '' Playfaiifs " axiom in the 
proof of this proposition, we proceed thus. 

To prove that the alternate angles AGff, v ^l 

GIfD are equal. ^^ ,^- "" p 

If they are not equal, draw another straight K- \ 

line I^L through G making the angle XGIf q Jv^ o 

equal to the angle GIfD. \ 

Then, since the angles KGIf, GIfD are equal, ^ 

JTZ is parallel to CD. [i. 27] 

Therefore ttvo sira^ht lines KL» AB intersecting at G are both parallel to 
the straight line CD : 

which is impossible (by the axiom). 

Therefore the angle ^C?^ cannot but be equal to the angle GHD. 

The rest of the proposition follows as in Euclid. 



I 



1.291 PROPOSITION 29 313 

Proof of Euclid's Postulate 5 from " Plasrfair's " axiom. 

Let ABy CD make with the transversal EF the angles AEF^ EFC 

together less than two right angles. 

To prove that AB^ CD meet towards A^ C p _— — -B 

Through E draw GH making with EF the angle Q.j^^;:::::;.^^-^^^^^^^— --H 

GJS/' equal (and alternate) to the angle EFD, ^ \ 

Thus G-^is parallel to CD, [i. 27] \ 

Then (i) AB must meet CD in one direction or ^ F 

the other. 

For, if it does not, AB must be parallel to CD\ hence we have two 

straight lines AB^ GH intersecting at E and both parallel to CD : 

which is impossible. ^j2^ v^la-^-^. juy^^ .^v^ 

Therefore AB^ CD must meet. 

(2) Since ABy CD meet, they must form a triangle with EF, 

But in any triangle any two angles are together less than two right angles. 

Therefore the angles AEF^ EFC (which are less than two right angles), 
and not the angles BEF^ EFD (which are together greater than two right 
angles, by i. 13), are the angles of the triangle ; 

that is, EAy FC meet in the direction of A^ C, or on the side of EF on 
which are the angles together less than two right angles. 

The usual course in modem text-books which use " Playfair's " axiom in 
lieu of Euclid's Postulate is apparently to prove i. 29 by means of the axiom, 
and then Euclid's Postulate by means of i. 29. 

De Morgan would introduce the proof of Postulate 5 by means of 
"Playfair's" axiom before i. 29, and would therefore apparently prove i. 29 as 
Euclid does, without any change. 

As between Euclid's Postulate 5 and " Playfair's " axiom, it would appear 
that the tendency in modem text-books is rather in favour of the latter. 
Thus, to take a few noteworthy foreign writers, we find that Rausenberger 
stands almost alone in using Euclid's Postulate, while Hilbert, Henrici and 
Treutlein, Rouch^ and De Comberousse, Enriques and Amaldi all use 
" Playfair's " axiom. 

Yet the case for preferring Euclid's Postulate is argued with some force by 

Dodgson (Euclid and his modem Rivals^ pp. 44—6). He maintains (i) that 

:' Playfair's" axiom in fact involves Euclid's Postulate, but at the same time 

I involves tnare than the latter, so that, to that extent, it is a needless strain on 

the faith of the learner. This is shown as follows. 

Given AB, CD making with EF\}^t, angles AEF, EFC together less than 
two right angles, draw GH through E so that the angles GEF, EFC are 
together equal to two right angles. 

Then, by i. 28, GH, CD are ''separational." 

We see then that any lines which have the property (a) that they make 
with a transversal angles less than two right angles have also the property {fi) 
that one of them intersects a straight line which is '' separational " from 
the other. 

Now Playfair's axiom asserts that the lines which have property (fi) meet 
if produced : for, if they did not, we should have two intersecting straight 
lines both '' separational " from a third, which is impossible. 

We then argue that lines having property (a) meet because lines having 
property (a) are lines having property (fi). But we do not know, until we 
have proved i. 29, that all pairs of lines having property (fi) have also property 



314 BOOK I [i. 29, 30 

(a). For anything we know to the contiary, class (fi) mmy be greater than 
class (a). Hence, if you assert anything of dass (/S), the logical ^ect is more 
extensive than if you assert it of dass (a) \ for you assert it, not only of that 
portion of class (fi) which is known to be included in dass (a), but abo of the 
unknown (but possibly existing) portion which is ff^/ so induded. < 

(2) Eudid's Postulate pjuU before the beginner clear and posUm con- 
ceptions, a pair of straight lines, a transversal, and two angl^ together less 
than two right angles, whereas " Playfair's " axiom requires him to reaEse a 
pair of straight lines which never meet though produced to infinity : a mgatioe 
conception which does not convey to the mmd any clear notion of therdative 
position of the lines. And (p. 68) Eudid's Postulate gives a direct criterion 
for judging that two straight hnes meet, a criterion which is constantly reqdied, 
e^. in I. 44. It is true that the Postulate can be deduced from ^Playftir's'' 
axiom, but editors frequently omit to deduce it, and then taddy assume it 
afterwards : which is the least justifiable course of all 



Proposition 30. 

Straight lines parallel to the same straight line are also 
parallel to one another. 

Let each of the straight lines AB^ CD be parallel to EF\ 
I say that AB is also parallel to CD. 
5 For let the straight line GK fall upon 
them. 

Then, since the straight line GK 
has fallen on the parallel straight lines 
AB, EF, 
o the angle AGK is equal to the 

angle GHF. [l 29] j 

Again, since the straight line GK has fallen on the parallel I 
straight lines EF, CD, - 

the angle GHF is equal to the angle GKD. [i. 29] 
5 But the angle AGK "w^s also proved equal to the angle 
GHF\ 

therefore the angle AGK is also equal to the angle 
GKD ; [C. N. il 

and they are alternate. 

o Therefore AB is parallel to CD. 

Q. E. D. 

«o. The usual conclusUn in general terms (" Therefore etc'*) repeatiiig the ennnciatioii 
is, curiously enough, wanting at the end of this proposition. 

The proposition is, as De Morgan points out, the logical equivalent of 
''Playfoir's" axiom. Thus, if X denote "pairs of straight lines intersecting one 




30. 3'] 



PROPOSITIONS 29—31 



3>S 







another," K^' pairs of straight lines parallel to one and the same straight line," 
we have 

No X is K, 
and it follows logically that 

No Y is X. 

De Morgan adds that a proposition is much wanted about parallels ^or 
perpendiculars) to two straight lines respectively making the same angles with 
one another as the latter do. The proposition may be enunciated thus : 

If the sides of one angle be respectively (i) parallel or (2) perpendicular to 
the sides of another ang^^ the two angles are either 
equal or supplementary. 

(i) Let DE be parallel to AB and GEF parallel 
to BC. 

To prove that the angles ABC^ DEG are equal 
and the angles ABC^ JDEF supplementary. 

Produce DE to meet BC in H, 

Then [i. 29] the angle DEG is equal to the angle 
DHC, 

and the angle ABC is equal to the angle DHC 
Therefore the angle DEG is equal to the angle ABC\ whence also the 
angle DEF is supplementary to the angle ABC 

(2) Let ED be perpendicular to AB, and GEF perpendicular to BC, 

To prove that the angles ABC, DEG are 
equal, and the angles ABC, DEF supplementary. 

Draw Ejy at right angles to ED on the side 
of it opposite to B, and (Uaw EG at right angles 
to EF on the side of it opposite to B, 

Then, since the angles BDE, DED, being 
right angles, are equal, 

ED is parallel to BA. [i. 27] 
Similarly EG' is parallel to BC 

Therefore [Part (i)] the angle DEG is equal to the angle ABC 
But, the right angle DED being equal to the right angle GEG, if the 
common angle GEuht subtracted, 

the angle DEG is equal to the angle DEG , 
Therefore the angle DEG is equal to the angle ABC\ and hence the 
angle DEF is supplementary to the angle ABC 



Proposition 31. 

Through a given point to draw a straight line parallel to a 
given straight line. 

Let A be the given point, and BC the given straight 
line; 
thos it is required to draw through the point A a straight 




line parallel to the straight line Bi 



3i6 BOOK I [i. 31, 3a 

Let a point D be taken at random on BC, and let AD be 
joined; on the straight line DA, 
and at the point A on it, let the ^ — 



angle DAE be constructed equal / 



to the angle ADC [i. 23] ; and let the 



straight 
straight 



ine AF be produced in a 
ine with EA. 



I 



Then, since the straight line AD falling on the two 
straight lines BC, EF has made the alternate angles EAD, 
ADC equal to one another, 

therefore EAF is parallel to BC. [i. «7] 

Therefore through the given point A the straight line 
EAF has been drawn parallel to the given straight line BC. 

Q. E, F, 

Proclus rightly remarks (p. 376, 14 — 20) that, as it is implied in i. 12 
that only one perpendicular can be drawn to a straight line from an external 
point, so here it is implied that only one straight line can be drawn through a 
point parallel to a given straight line. The construction, be it observed, 
depends only upon i. 27, and might therefore have come directly after that 
proposition, ^^y then did Eucbd postpone it until after i. 29 and L 30? 
Presumably because he considered it nec^sary, before giving the constructioii, 
to place beyond all doubt the fact that only one such parallel am be drawn. 
Proclus infers this fact from i. 30 ; for, he says, if two straight lines could be 
drawn through one and the same point parallel to the same straight line, the two 
straight lines would \^ parallel, though intersecting at the given point : which 
is impossible. I think it is a fair inference that Euclid would have considered 
it necessary to justify the assumption that only one parallel can be drawn 
by some such argument, and that he deliberately determined that his own 
assumption was more appropriate to be made the subject of a Postulate 
than tfie assumption of the uniqueness of the parallel 



Proposition 32. 

In any triangle, if one of the sides be produced, the exterior 
angle is equal to the two interior and opposite angles^ and the 
three interior angles of the triangle are equal to two right 
angles. 

Let ABC be a triangle, and let one side of it BC be 
produced to D ; 

I say that the exterior angle ACD is equal to the two 
interior and opposite angles CAB, ABC, and the three 
interior angles of the triangle ABC^ BCA, CAB are equal 
to two right angles. 




I, 32] PROPOSITIONS 31. 32 317 

For let CE be drawn through the point C parallel to the 
straight line AB. [i. 31] 

Then, since AB is parallel to CE, 
and AC has fallen upon them, 
the alternate angles BAC, ACE are 
equal to one another. [i. 29] 

Again, since AB is parallel to 
CE, 

and the straight line BD has fallen upon them, 

the exterior angle ECD is equal to the interior and opposite 
angle ABC. [i. 29] 

But the angle ACE was also proved equal to the angle 
BAC\ 

therefore the whole angle ACD is equal to the two 
interior and opposite angles BAC, ABC 

Let the angle ACB be added to each ; 

therefore the angles ACD, ACB are equal to the three 
angles ABC, BCA, CAB. 

But the angles ACD, ACB are equal to two right angles; 

P- 13] 
therefore the angles ABC, BCA, CAB are also equal 

to two right angles. 

Therefore etc, 

Q. E. D. 

This theorem was discovered in the very early stages of Greek geometry. 
What we know of the history of it is gathered from three allusions found in 
Eutocius, Proclus and Diogenes Laertius respectively. 

I. Eutocius at the beginning of his commentary on the Conies of 
''Apollonius (ed. Heiberg, Vol. 11. p. 170) quotes Geminus as saying that *'the 
ancients (oc ^x^^ot) investigated the theorem of the two right angles in each 
individual species of triangle, first in the equilateral, again in the isosceles, 
and afterwards in the scalene triangle, and later geometers demonstrated the 
general theorem to the effect that in any triangle the three interior angles are 
equal to two right angles." 

2. Now, according to Proclus (p. 379, 2 — 5), Eudemus the Peripatetic 
refers the discovery of this theorem to the Pythagoreans and gives what he 
affirms to be their demonstration of it This demonstration will be given 
below, but it should be remarked that it is general, and therefore that the 
"later geometers" spoken of by Geminus were presumably the Pythagoreans, 
whence it appears that the ''ancients" contrasted with them must have 
belonged to the time of Thales, if they were not his Egyptian instructors. 

3. That the truth of the theorem was known to Thales might also 
be inferred from the statement of Pamphile (quoted by Diogenes Laertius, 
L 24 — 5, p. 6, ed. Cobet) that "he, having learnt geometry from the 



3i8 BOOK I [L39 

Egyptians, was the first to inscribe a right-angled triang^ in a aide and 
sacrificed an ox" (on the strength of it) ', in odier words, he discovered that 
the angle in a semicircle is a right angle. No doubt, when this fact wu once 
discovered (empinca/fy^ say), the consideration of the two isosceles triangles 
having the centre for vertex and the sides of the right angle for bases 
respectively, with the help of the theorem of EucL i. 5, also known to 
Thales, would easily lead to the conclusion that the sum of the angles of 
a right-angled triangle is equal to two right angles, and it could be readfly 
inferred that the angles of any triangle were likewise equal to two right angles 
(by resolving it into two right-angled triangles). But it is not easy to see how 
the property of the angle in a semicircle could h^prwed except (in the revise 
order) by means of the equality of the sum of the angles of a right-an^ii 
triangle to two right angles ; and hence it is most natural to suppose, with 
Cantor, that Thales proved it (if he did prove it) practically as Euclid does 
in III. 31, i.e. by means of i. 32 as applied to righi-angled triangles at all events. 
If the theorem of i. 32 was proved before Thales' time, or by Thales 
himself, by the stages indicated in the note of Geminus, we may be satisfied 
that the reconstruction of the argument of the older proof by Hankd 
(pp. 96 — 7) and Cantor (i„ pp. 143 — 4) is not far wronj;. First, it must have 
be^sn observed that six angles equal to an angle of an eqmlateral triangle would, 
if placed adjacent to one another round a common vertex, fill up the whde 
space round that vertex. It is true that Produs attributes to the Pythafpreans 
the general theorem that only three kinds of regular polygons, the equilateral 
trianf^le, the square and the regular hexagon, can fill up the entire space round 
a point, but the practical Imowled^ge that equilateral triangles have this 
property could hardly have escaped the Egyptians, whether they made floors 
with tiles in the form of equilateral triangles or regular hexagons (Allman, 
Greek Geometry from Thales to Euciid, p. 13) or joined the ends of adjacent 
radii of a figure like the six-spoked wheel, which was their common form of 
whed fi'om the time of Ramses II. of the nineteenth Dynasty, say 1300 b.c 
(Cantor, i,, p. 109). It would then be dear that six angles equal to an angle 
of an equilateral triangle are equal to four right angles, and therefore that the 
three angles of an equilateral triangle are equal to two right angles. (It would 
be as clear or clearer, from observation of a square divided into two triangles 
by a diagonal, that an isosceles right-angled triangle has each of its equal 
angles equal to half a right angle, so that an isosceles right-angled triangle 
must have the sum of its angles equal to two right angles.) Next, with leffxA 
to the equilateral triangle, it could not fail to be observed 
that, if AD were drawn from the vertex A perpendicular 
to the base BC, each of the two right-angled triangles so 
formed would have the sum of its angles equal to two right y 

angles ; and this would be confirmed by completing the / 

rectanp;le ADCE, when it would be, seen that the rectangle / 

(with Its angles equal to four right angles) was divided by / 

its diagonal into two equal triangles, each of which had 6 
the sum of its angles equal to two right angles. Next it 
would be inferred, as the result of drawing the diagonal of any rectangle and 
observing the equality of the triangles forming the two halves, that the sum of 
the angles of any right-angled triangle is equal to two right angles, and henoe 
(the two congruent right-angled triangles being then pla^ so as to form one 
isoscdes triangle) that the same is true of any isosceles triangle. Only the 
last step remained, namely that of observing that any triangle could be 
regurded as the half of a rectangle (drawn as indicated in the next figure), or 



L32] PROPOSITION 32 319 

simply that any triangle could be divided into two right-angled triangles, 

I whence it would be inferred that in general the 

I sum of the angles of any triangle is equal to two f --^^^p;— • ; 

I right angles. j X \ ^s^^^ j 

Such would be the probabilities if we could \ X \. i 

absolutely rely upon the statements attributed to \,^__ ! ^J 

Pamphile and Geminus respectively. But in fact 

there is considerable ground for doubt in both cases. 

1. Pamphile's story of the sacrifice of an ox by Thales for joy at his 
discovery that the angle in a semicircle is a right angle is too suspiciously like 
the similar story told with reference to Pythagoras and his discovery of the 
theorem of Eucl. i. 47 (Proclus, p. 426, 6 — 9). And, as if this were not 
enough, Diogenes Laertius immediately adds that '* others, among whom is 
Apollodorus the calculator (d XoytorcKo^), say it was Pythagoras" (sc. who 
*' mscribed the right-angled triangle in a circle "). Now Pamphile lived in the 
reign of Nero (a.d. 54 — 68) and therefore some 700 years after the birth of 
Thales (about 640 b.c). I do not know on what Max Schmidt bases his 
statement {Kulturhistorische Beitrdge%ur Kenntnis des griechischen und romischm 
AUertums^ 1906, p. 31) that "other, much older^ sources name Pythagoras as 
the discoverer of the said proposition," because nothing more seems to be 
known of Apollodorus than what is stated here by Diogenes Laertius. But it 
would at least appear that Apollodorus was only one of several authorities 
who attributed the proposition to Pythagoras, while Pamphile is alone 
mentioned as referring it to Thales. Again, the connexion of Pythagoras with 
the investigation of the right-angled triangle makes it a priori more likely 
that it would be he who would discover its relation to a semicircle On 
the whole, therefore, the attribution to Thales would seem to be more than 
doubtful. 

2. As regards Geminus' account of the three stages through which the 
proof of the theorem of 1. 32 passed, we note, first, that it is certainly not 
confirmed by Eudemus, who referred to the Pythagoreans the discovery of the 
theorem that the sum of the angles of any triangle is equal to two right 
angles and says nothing about any gradual stages by which it was proved. 
Secondly, it must be admitted, I think, that in the evolution of the proof as 
reconstructed by Hankel the middle stage is rather artificial and unnecessary, 

^since, once it is proved that any right-angled triangle has the sum of its angles 
ecjual to two ri^ht angles, it is just as easy to pass at once to any scalene 
triangle (which is decomposable into two unequal right-angled triangles) as to 
the isosceles triangle made up of two congruent right-angled triangles. Thirdly, 
as Heiberg has recently pointed out (Mathematisches zu Aristoteles^ p. 20), it 
is quite possible that the statement of Geminus from beginning to end is 
simply due to a misapprehension of a passage of Aristotle {Anal, Post. i. 5, 
74 a 25). Aristotle is illustrating his contention that a property is not 
scientifically proved to belong to a class of things unless it is proved to belong 
primarily {wfHarov) and generally (iraMXou) to the whole of the class. His first 
illustration relates to parallels making with a transversal angles on the same 
side together equal to two right angles, and has been quoted above in the note 
on I. 27 (pp. 308—9). His second illustration refers to the transformation of 
a proportion altemando^ which (he says) ''used at one time to be proved 
separately " for numbers, lines, solids, and times, although it admits of being 
proved of all at once by one demonstration. The third illustration is : " For 
the same reason, even if one should prove (ovS* av rvi Sciif^) with reference to 







320 BOOK I [1.32 

each (sort of) triangle, the equilateral, scalene and isosceles, separateljr, that 
each has its angles equal to two right angles, either by one proof or by different 
proofs, he does not yet know that t?u trian^^ i.e. the triangle in gtmrai^ has 
Its angles equal to two right angles, except in a sophistical sense, even though 
there exists no triangle other than triangles of the kinds mentioned. For he 
knows it, not qu& triangle, nor of evtry triansle, except in a numerical sense 
(car apiO/ior); he does not know it noHonally (icar cISo«) of every triangle^ even 
though there be actually no triangle which he does not know." 

The difference between the phrase *' used at one time to be proved " in 
the second illustration and '' if any one should prove " in the third appears to I 
indicate that, while the former referred to a historical fact, the latter does not; 
the reference to a person who should prove the theorem of i. 32 for the three 
kinds of triangle separately, and then claim that he had proved it generally, i| 
states a purely hypothetical case, a mere illustration. Yet, coming after the j 
historical Ibxx stated in the preceding illustration, it might not unnaturally give 
the impression, at first sight, that it was historicsil too. 

On the whole, therefore, it would seem that we cannot safely go behind 
the dictum of Eudemus that the discovery and proof of the theorem of i. 32 
in all its generality were Pythagorean. This does not however preclude its 
having been discovered by stages such as those above set out after Hankel 
and Cantor. Nor need it be doubted that Thales and even his Egyptian 
instructors had advanced some way on the same road, so far at all events as 
to see that in an equilateral triangle, and in an isosceles right-angled triangle, 
the sum of the angles is equal to two right angles. 

The Pythagorean proof. 

This proof, handed down by Eudemus (Proclus, p. 379, 2 — 15), is no 
elegant than that given by EucUd, and is a natural 
development from the last figure in the recon- 
structed argument of Hankel. It would be seen, 
after the theory of parallels was added to geometry, 
that the actual drawing of the perpendicular and 
the complete rectangle on BC as base was 
unnecessary, and that the parallel to BC through 
A was all that was required. 

Let ABC be a triangle, and through A draw DE parallel to BC [1. 31] 

Then, since BC^ DE are parallel, 
the alternate angles DAB^ ABCzit equal, [i. 29] 

and so are the alternate angles EAd ACB also. 

Therefore the angles ABC^ ACB are together equal to the angles 
DAB, EAC 

Add to each the angle BAC; 
therefore the sum of the angles ABC, ACB, BAC is equal to the sum of the 
angles DAB, BAC, CAE, that is, to two right angles. 

Euclid's proof pre-Euclidean. 

The theorem of i. 32 is Aristotle's favourite illustration when he wishes to 
refer to some truth generally acknowledged, and so often does it occur that 
it is often indicated by two or three words in themselves hardly intelli^ble, 
e.g. TO fivcrlv ^p^ois (Ana/, Post, i. 24, 85 b 5) and vwapxti wavrl rpiyiiytf ro fivo 
{Md, 85 b 11). 

One passage (Metaph. 1051 a 24) makes it clear, as Heiberg (op,*dt. 




I. 32] PROPOSITION 32 321 

p. 19) acutely observes, that in the proof as Aristotle knew it Euclid's 
construction was used. "Why does the triangle make up two right angles? 
Because the angles about one point are equal to two right angles. If then the 
parallel to the side had been drawn up (avrJKro), the fact would at once have 
oeen clear from merely looking at the figure." The words "the angles about 
one point" would equally fit the Pythagorean construction, but "drawn 
upwards " applied to the parallel to a side can only indicate Euclid's. 

Attempts at proof independently of parallels. 
The most indefatigable worker on these lines was Legendre, and a sketch 
of his work has been given in the note on Postulate 5 above. 

One other attempted proof needs to be mentioned here because it has 
^ found much favour. I allude to 

Thibaut's method. 

This appeared in Thibaut's Grundriss der reinen Mathematik, Gottingen 
(2 ed. 1809, 3 ed. 181 8), and is to the following effect 

Suppose CB produced to D^ and let BD (produced to any necessary extent 
either way) revolve in one direction (say 
clockwise) first about B into the position 
BA^ then about A into the position oi AC 
produced both ways, and lastly about C 
mto the position CB produced both ways. 

The argument then is that the straight 
line BD has revolved through the sum of 
the three exterior angles of the triangle. 
But, since it has at the end of the revolution 

assumed a position in the same straight line with its original position, it must 
have revolved through four right angles. 

Therefore the sum of the three exterior angles is equal to four right 
angles; 

frx)m which it follows that the sum of the three angles of the triangle is equal 
to two right angles. 

But it is to be observed that the straight line BD revolves about different 
points in it^ so that there is translation combined with rotatory motion, and it 
is necessary to assume as an axiom that the two motions are independent, and 
^therefore that the translation may be neglected. 

Schumacher (letter to Gauss of 3 May, 1831) tried to represent the 
rotatory motion graphically in a second figure as mere motion round a point ; 
but Gauss (letter of 17 May, 1831) pointed out in reply that he really 
assumed, without proving it, a proposition to the effect that " If two straight 
lines (i) and (2) which cut one another make angles A\ A" with a straight 
line (3) cutting both of them, and if a straight line (4) in the same plane is 
I likewise cut by (i^ at an angle A\ then (4) will be cut by (2) at the angle A\ 
But this proposition not only needs proof, but we may say that it is, in 
I essence, the very proposition to be proved" (see Engel and Stackel, Die 
Theorie der ParaUelHnien von Euklid bis auf Gauss^ 1895, P* ^3^)* 

How easy it is to be deluded in this way is plainly shown by Proclus' 
attempt on the same lines. He says (p. 384, 13—21) that the truth of the 
theorem is borne in upon us by the help of " common notions " only. " For, 
if we conceive a straight line with two perpendiculars drawn to it at its ex- 
tremities, and if we then suppose the perpendiculars to (revolve about 
their feet and) approach one another, so as to form a triangle, we see that, 

H. E. 21 




322 BOOK I [i. 32, 33 

to the extent to which they converge, they diminish the right an^es wtidb, 
they made with the straight line, so that the amomit taken from the rifjtxt 
angles is also the amount added to the vertical angle of the- triangle, and the 
three angles are necessarily made ec^ual to two right angles." But a moment^* I 
reflection shows that, so far from bemg founded on mere " common notiooa," 
the supposed proof assumes, to begin with, that, if the perpendiculars ap- 
proach one another ever so litde, they will then form a triangle immediately, 
i.e., it assumes Postulate 5 itself; and the fact about the vertical aiigle can only 
be seen by means of the equality of the alternate angles exhibited by drawing 
a perpendicular from the vertex of the triangle to the base, Le. KparaiMto 
eitner of the original perpendiculars. 

Extension to polygons. 

The two important corollaries added to i. 32 in Simson's edition are given 
by Proclus ; but Proclus' proof of the first is different from, and pertiaps 
somewhat simpler than, Simson's. 

1. The sum of the interior anf^lts of a convex reetilifieai figure is equal to 
twice as many right angles as the figure has sides, 
less four. 

For let one angular point A be joined to all 
the other angular points with which it is not con- 
nected already. 

The figure is then divided into triangles, and 
mere inspection shows 

(i) that the number of triangles is two less 
than the number of sides in the figure, 

(2^ that the sum of the angles of all the ^ 

triangles is equal to the sum of sdl the interior angles of the fi^re. | 

Smce then the sum of the angles of each triangle is equal to two right angles, 
the sum of the interior angles of the figure is equal to 2 (/r-2) right angles, I 
Le. {2n - 4) right angles, where n is the number of sides in the figure. \ 

2. The exterior angles of any convex rectilineal 
figure are together equal to four right cutgles. 

For the interior and exterior angles together are 
equal to 2n right angles, where n is the numb^ of sides. 

And the intenor angles are together equal to 
(2ff-4) right angles. 

Therefore the exterior angles are together equal to 
(bur ri^ht angles. 

This last property is already quoted by Aristotle 
as true of all rectilineal figures in two passages (Anal, 
Ast. I. 24, 85 b 38 and 11. 17, 99 a 19). 

Proposition 33. 
TAe stra^kt iines joining equal ami parallel straight 
lines (at the extremities which are) in the same directions 
{respectively) are themselves also equcU and parallel. 

Let AB, CD be equal and parallel and let the straight 
5 lines AC^ BD join them (at the extremities which are) in the 
same directions (respectively) ; 





I. 33i 34] PROPOSITIONS 32—34 323 

; I say that AC, BD are also equal and parallel. 
Let BC be joined. 
Then, since AB is parallel to CD, 
10 and BC has fallen upon them, 

the alternate angles ABC, BCD 
are equal to one another. [i. 29] 

And, since AB is equal to CD, 
and BC is common, 
15 the two sides AB, BC are equal to the two sides DC, CB ; 
and the angle ABC is equal to the angle BCD ; 

therefore the base -^C is equal to the base BD, 
and the triangle ABC is equal to the triangle DCB, 
and the remaining apgles will be equal to the remaining angles 
20 respectively, namely those which the equal sides subtend ; [1. 4] 
therefore the angle ACB\% equal to the angle CBD. 
And, since the straight line BC falling on the two straight 
lines AC, BD has made the alternate angles equal to one 
another, 
25 ^C is parallel to BD. [i. 27] 

And it was also proved equal to it. 
Therefore etc. q. e. d. 

I. joining... (at the extremities which are) in the same directions (respectively). 
I have for clearness' sake inserted the words in brackets though they are not in Uie original 
Greek, which has "joining... in the same directions" or "on the same sides,*' heX rd odrd iiipf^ 
iwi^€vyw6ciuffai. The expression "towards the same parts,'* though usage has sanctioned it, 
b perhaps not quite satisfiurtory. 

15. DC, CB and 18. DCB. The Greek has **BC, CD*' and " BCD" in these places 
respectively. Euclid is not always careful to write in corresponding order the letters denoting 
corresponding points in congruent figures. On the contrary, he evidently prefers the alpha- 
betical order, and seems to disdain to alter it for the sake of bqginners or others who might 
be confused by it. In the case of angles alteration is perhaps unnecessary ; but in the case 
^ of trianeles and pairs of corresponding sides I have ventured to alter the order to that which 
the mathematician of to-day expects. 

This proposition is, as Proclus says (p. 385, 5), the connecting link between 
the exposition of the theory of parallels and the investigation of parallelograms. 
For, while it only speaks of equal and parallel straight lines connecting those 
ends of equal and parallel straight lines which are in the same directions, it 
gives, without expressing the fact, the construction or origin of the parallelogram, 
so that in the next proposition Euclid is able to speak of " parallel6grammic 
areas" without any further explanation. 

Proposition 34. 

In parallelogrammic areas the opposite sides and angles 
are equal to one another, and the diameter bisects the areas. 

Let ACDB be a paFallelogrammic area, and BC its 
diameter ; 

21 — 2 




324 BOOK I [i. 34 

5 I say that the opposite sides and angles of the parallelogram 
ACDB are equal to one another, and the diameter BC 
bisects it 

For, since AB is parallel to CD, 
and the straight line BC has fallen 
ioupon them, 

the alternate angles ABC, BCD 
are equal to one another. [i. 29] 

Again, since -<4C is parallel to BD, 
and BC has fallen upon them, 

15 the alternate angles ACB, CBD are equal to one 

another. [l 29] 

Therefore ABC, DCB are two triangles having the two 
angles ABC, BCA equal to the two angles DCB, CBD 
respectively, and one side equal to one side, namely that 
20 adjoining the equal angles and common to both of them, BC\ 
therefore they will also have the remaining sides equal 
to the remaining sides respectively, and the remaining angle 
to the remaining angle ; [i. 26] 

therefore the side AB is equal to CD, 
25 and^Cto^A 

and further the angle BAC is equal to the angle CDB. 
And, since the angle ABC is equal to the angle BCD, 
and the angle CBD to the angle ACB, 

the whole an^^le ABD is equal to the whole angle A CD. 

[C.N.2\ 

30 And the angle BAC^^s also proved equal to the angle CDB. ^ 
Therefore in parallelogrammic areas the opposite sides 
and angles are equal to one another. 

I say, next, that the diameter also bisects the areas. 
For, since AB is equal to CD, 
35 and BC is common, 
the two sides AB, BC are equal to the two sides DC, CB 
respectively ; 

and the angle ABC is equal to the angle BCD ; 
therefore the base AC is also equal to DB, 
40 and the triangle ABC is equal to the triangle DCB. [i. 4] 
Therefore the diameter BC bisects the parallelogram 

ACDB. Q. E. D. 



1.34] PROPOSITION 34 325 

I. It b to be observed that, when parallelograms have to be mentioned for the first time, 
Euclid calls them *'parallelogrammtc areas*' or, more exactly, ''parallelogram*' areas 
{wapaXKii\6yp€ifA/ia x^^pfa). The meaning is simply areas bounded bv parallel straight lines 
with the further limitation placed upon the term by Euclid that oiAy jeur-Hded figures are so 
called, although of course there are certain regular polygons which have opposite sides 
parallel, and which therefore might be said to be areas bouiMed by parallel straight lines. We 

Sther from Proclus (p. 30^) that the word "parallelogram** was first introduced by Euclid, 
It its use was suggestea by i. 33, and that the formation of the word «-a/>aXXiyX(&Y/N^(/Aot 
(paralleMined) was analogous to that of M^rfpanitw (straight-lined or rectilineal). 

17, 18, 40. DCB and 36. DC, CB. The Greek has in these places ** BCD'* and 
••C/>, BC respectively. Cf. note on i. 33, lines 15, 18. 

After specif^ng the particular kinds of parallelograms (squares and rhombi) 
in which the diagonals bisect the angles which they join, as well as the areas, 
and those (rectangles and rhomboids) in which the diagonals do not bisect 
the angles, Proclus proceeds (pp. 390 sqq.) to analyse this proposition with 
reference to the distinction in Aristotle's AncU. Post (i. 4, 5, 73 a 21 — 74 b 4) 
between attributes which are only predicable of every individual thing (icara 
iroi^os) in a class and those which are true of \i primarily {tovtov xpc^ov) and 
genercUly (ko^oXov). We are apt, says Aristotle, to mistake a proof «aTa 
vavros for a proof rovrov xpcorov KoBokxn) because it is either impossible to 
find a higher generality to comprehend all the particulars of which the 
predicate is true, or to find a name for it (Part of this passage of Aristotle 
has been quoted above in the note on i. 32, pp. 319 — 320.) 

Now, says Proclus, adapting Aristotle's distinction to theorems^ the present 
proposition exhibits the distinction between theorems which are general and 
theorems which are not general. According to Proclus, the first part of 
the proposition stating that the opposite sides and angles of a parallelogram 
are equal is general because the property is only true of parallelograms ; but 
the second part which asserts that the diameter bisects the area is not general 
because it does not include all the figures of which this property is true, e.g. 
circles and ellipses. Indeed, says Proclus, the first attempts upon problems 
seem usually to have been of this partial character (/Acpucoircpai), and generality 
was only attained by degrees. Thus ''the ancients, after investigating the 
fact that the diameter bisects an ellipse, a circle, and a parallelogram 
respectively, proceeded to investigate what was common to these cases," 
though " it is difficult to show what is common to an ellipse, a circle and a 
parallelogram." 

I doubt whether the supposed distinction between the two parts of the 
proposition, in point of " generality," can be sustained. Proclus himself admits 
that it is presupposed that the subject of the proposition is a quadrilateral^ 
because there are other figures (e.g. regular polygons of an even number of 
sides) besides parallelograms which have their opposite sides and angles 
equal; therefore the second part of the theorem is, m this respect^ no more 
< general than the other, and, if we are entitled to the tacit limitation of the 
1 theorem to quadrilaterals in one part, we are equally entitled to it in the other. 

It would almost appear as though Proclus had drawn, the distinction for 
the mere purpose of alluding to investigations by Greek geometers on the 
general subject of diameters of all sorts of figures; and it may have been these 
which brought the subject to the point at which Apollonius could say in the 
first definitions at the beginning of his Conies that '* In any bent line^ such as 
is in one plane, I give the name diameter to any straight line which, being 
drawn from the bent line, bisects all the straight lines (chords) drawn in the 
line parallel to any straight line.** The term bent line {KOfLwvktf ypa^'q) 
includes, e.g. in Archimedes, not only curves, but any composite line made 



3«6 



BOOK I 



[>• 34, 35 




up of straight lines and curves joined together in any manner. It is of course 
clear that either diagonal of a parallelogram bisects all lines drawn within the 
parallelogram parallel to the omer diagonal 

An-Nairizi gives after i. 31 a neat construction for dividing a straight line 
into any number of equal parts (ed. Curtze, p. 74, ed. B^thom-Heibeig, 
pp. 141 — 3) which requires only one measuranent repeated, together with the 
properties of parallel lines including i. 33, 34. As i. 33, 34 are assumed, I 
place the problem here. The particular case taken is the problem of dividing 
a straight line into three equal parts. 

Let AB be the given straight line. Draw AC^ BD at right angles to it 
on opposite sides. 

An-Nairizi takes AC^ BD of the same 
length and then bisects AC at E and BD 
at F, But of course it is even simpler to 
measure AE^ EC along one perpendicular 
equal and of any length, and BF^ FD along 
the other also equal and of the same length. 

Join ED, CF meeting AB \n G^ H 
respectively. 

Then shall AG, GH, iET^ all be equal 

Draw HK pandlel to AC, or at right 
angles to AB. 

Since now EC, FD are equal and parallel 
ED, CFiJt equal and parallel. [i- 33 J 

And HK was drawn parallel to AC. 

Therefore ECHK is a parallelogram ; whence KH is equal as well as 
parallel to EC, and therefore to EA. 

The triangles EAG, KHG have now two angles respectively equal and the 
sides AE, /^ equal 

Thus the triangles are equal in all respects, and 
AG is equal to GH. 

Similarly the triangles KHG, FBH are equal in all respects, and 
GH is equal to HB. 

If now we wish to extend the problem to the case where AB is to be 
divided into n parts, we have only to measure (n—\) successive equal lengths 
along AC and {n-i) successive lengths, each equal to the others, along BD. 
Then join the first point arrived at' on AC to the last point on BD, the 
second on ^^C to the last but one on BD, and so on ; and the joining lines 
cut AB in points dividing it into n equal piauts. 

Proposition 35. 

Parallelograms which are on the same base and in the 
same parallels are equal to one another. 

Let ABCD, EBCF be parallelograms on the same base 
BC and in the same parallels AF, BC ; 
5 I say that ABCD is equal to the parallelogram EBCF. 
For, since ABCD is a parallelogram, 

AD is equal to BC. [l 34] 




I. 35] PROPOSITIONS 34, 35 3^7 

I For the same reason also 

I EF is equal to BCy 

iio so that AD is also equal to EF\ \C. N, i] 

and DE is common ; 

! therefore the whole AE is equal to the whole Z?/^ 

[C. N. 2] 

But AB is also equal to DC\ [i. 34] 

therefore the two sides EA, AB are equal to the two sides 

}SED, DC respectively, 

and the angle FDC is equal to 
the angle EAB, the exterior to the 
interior ; [i. 29] 

therefore the base EB is equal 
20 to the base FQ 

and the triangle EAB will be equal to the triangle FDC. 

[1.4] 
Let DGE be subtracted from each ; 

therefore the trapezium ABGD which remains is equal to 
the trapezium EGCF "which remains. [C N. 3] 

25 Let the triangle GBC be added to each ; 
therefore the whole parallelogram A BCD is equal to the whole 
parallelogram EBCF. [C N. 2] 

Therefore etc. 

Q. E. D. 

11. FDC. The text has "/>/-C." 

33. Let DOE be subtracted. Euclid speaks of the triangle DGE without any 
explanation that, in the case which he takes (where AD^ EF have no point in common), 
.^ B£t CD must meet at a point G between the two parallels. He allows tnis to appear from 
the figure simply. 

Equality in a new sense. 

It is important to observe that we are in this proposition introduced for 
the first time to a new conception of equality between figures. Hitherto we 
have had equality in the sense of congruence only, as applied to straight lines, 
angles, and even triangles (cf. i. 4). Now, without any explicit reference to 
any change in the meaning of the term, figures are inferred to be equal which 
are equal in area or in content but need not be of the same farm. No 
definition of equality is anywhere given by Euclid ; we are left to infer its 
meaning from the few axioms about " equal things." It will be observed that 
in the above proof the " equality " of two parallelograms on the same base 
and between the same parallels is inferred by the successive steps (i) of 
subtracting one and the same area (the triangle DGE) from two areas equal 
in the sense of congruence (the triangles AEB^ ^PC\ and inferring that the 
remainders (the trapezia ABGD^ EGCF) are "equal"; (a) of addi^ one and 



328 BOOK I [i. 3S 

the same area (the triangle GBC) to each of the latter ** equal" trapezia, and 
inferring the equality of the respective sums (the two given parallelograms). 

As is well known, Simson (after Qairaut) slightly altered the proof in order 
to make it applicable to all the three possible cases. The alteration 
substituted one step of subtracting congruent areas (the triangles AEB^ DFC) 
from one and the same area (the trapezium ABCF) for the iu>o steps above 
shown of first subtracting and then adding a certain area. 

While, in either case, nothing more is explicitly used than the axioms that, 
if equals be added to equals^ the wholes are equal and that, if equals be subtracted 
from equals^ the remainders are equals there is the further tacit assumption that 
it is indifferent to what part or from what part of the same or equal areas the 
same or equal areas are added or subtracted. De Morgan observes that the 
postulate "an area taken from an area leaves the same area from whatever 
part it may be taken " is particularly important as the key to equality of non- 
rectilineal areas which could not be cut into coincidence geometrically. 

Legendre introduced the word equivalent to express this wider sense of 
equality, restricting the term equal to things equal in the sense of congruent ; 
and this distinction has been found convenient. 

I do not think it necessary, nor have I the space, to give any account of 
the recent developments of the theory of equivalence on new lines represented 
by the researches of W. Bolyai, Duhamel, ])e Zolt, Stolz, Schur, Veronese, 
Hilbert and others, and must refer the reader to Ugo Amaldi's article Sulla 
teoria dell' equivalenza in Questiani riguardanti la geometria eUmentare 
(Bologna, 1900), pp. 103 — 142, and to Max Simon, Ober die Entwicklung der 
Elcmentar-^ometrie im XIX. fahrhumUrt {Ijtvpiagy 1906), pp. 115 — lao, with 
their full references to the literature of the subject I may however rder to 
the suggestive distinction of phraseology used by Hilbert (Grundlagen der 
Geometrie, pp. 39, 40) : 

(i) "Two polygons are called divisibly-equal (zerlegungsgleich) if they can 
be divided into z, finite number of triangles which are congruent two and two." 

(2) "Two polygons are called equal in content {inhaltsgleich) or of equal 
content if it is possible to add divisibly-equal polygons to them in such a way 
that the two combined polygons are ditnsibly-equaiy 

(Amaldi suggests as alternatives for the terms in (i) and (2) the expressions 
equivalent by sum and equivalent by difference respectively.) 

From these definitions it follows that " by combining divisibly-equal ' 
polygons we again arrive at divisibly-equal polygons; and, if we subtract 
divisibly-equal polygons from divisibly-equal polygons, the polygons remaining 
are equal in content" 

The proposition also follows without difficulty that, "if two polygons are 
divisibly-equal to a third polygon, they are also divisibly-equal to one another ; 
and, if two polygons are equal in content to a third polygon, they are equal in 
content to one another." 

The difTerent cases. 

As usual, Proclus (pp. 399 — 400), observing that Euclid has given only the 
most difficult of the three possible cases, adds the other two with separate 
proofs. In the case where E ii) the figure of the proposition falls between A 
and A he adds the congruent triangles ABE^ DCF respectively to the 
smaller trapezium EBCD^ instead of subtracting them (as Simson does) from 
the larger trapezium ABCF, 



1.35] PROPOSITION 35 339 

An ancient ** Budget of Paradoxes." 

Proclus observes (p. 396, 12 sqq.) that the present theorem and the 
similar one relating to triangles are among the so^^alled iMiradoxical theorems 
of mathematics, since the uninstructed might well regard it as impossible that 
the area of the parallelograms should remain the same while the length of the 
sides other than the base and the side opposite to it may increase indefinitely. 
He adds that mathematicians had made a collection of such paradoxes, the 
so-called treasury of paradoxes (6 irofxiSo^os rimoi) — cf. the similar expressions 
fifWQii dvaXv6fi€voi (treasury of analysis) and rotros d/(rrpovofMVfitvo9 — in the same 
way as the Stoics with their illustrations {Zavtp ol awo rSj$ Sroas hrl rmv 
&cy/iar«t>v). It may be that this treasury of paradoxes was the work of 
Erycinus quoted by Pappus (iii. p. 107, 8) and mentioned above (note on 
I. 31, p. 290). 

Locus-theorems and loci in Greek geometry. 

The proposition i. 35 is, says Proclus (pp. 394 — 6), the first locus-theorem 
(rorucov ^coipi^/ia) given by Euclid. Accordingly it is in his note on this 
proposition that Proclus gives us his view of the nature of a locus-theorem 
and of the meaning of the word locus (roVos) ; and great importance attaches 
to his words because he is one of the three writers (Pappus and Eutocius 
being the two others) upon whom we have to rely for all that is known of the 
Gredc conception of geometrical loci. 

Proclus' explanation (pp. 304, 15 — 395, 2) is as follows. "I call those 
(theorems) locus-theorems (roimra) in which the same property is found to exist 
on the whole of some locus (irpoc oh^ rm, r6iw^\ and (I call) a locus a position 
of a line or a surface producing one and the same property (ypofifi^ 1j iwi- 
^vtCoLS $€<nv voiovcrav Iv icai ravrov crv/utima/ia). *For, of locus-theorems, some 
are constructed on lines and others on siirfaces (rcSv yap rmnKC^v r& /ack k(m 
wph% ypafjLfjLOii (rvFurra/uicva, ra Sc irpoc ciri^avciatf ). And, since some lines are 
plane (cirivcSoi) and others solid (arcpcoi)— those being plane which are simply 
conceived of in a plane («Sv cv mirc&p 6,irkfj 17 vArftm), and those solid the 
origin of which is revealed from some section of a solid figure, as the cylin- 
drical helix and the conic lines (cos rifs tcvXivSpucff^ IXucoi koI ron^ KWfucQi^ 
ypafifjLwv) — I should say (^rfv av) further that, of locus-theorems on lines, 
. some give a plane locus and others a solid locus." 

Leaving out of sight for the moment the class of loci on surfaces^ we find 
that the distinction between plane and solid loci^ or plane and solid lineSy was 
similarly understood by Eutocius, who says (Apollonius, ed. Heiberg, 11. 
p. 184) that ^^ solid loci have obtained their name from the fact that the lines 
used in the solution of problems regarding them have their origin in the 
section of solids, for example the sections of the cone and several others.** 
Similarly we gather from Pappus that plane loci were straight lines and circles, 
and solid loci were conies. Thus he tells us (vii. p. 672,' 20) that Aristaeus 
wrote five books of Solid Loci '' supplementary to (literally, continuous with) 
the conies"; and, though Hultsch brackets the passage (vii. p. 662, 10 — 15) 
which says plainly that plane loci are straight lines and circles, while solid loci 
are sections of cones, i.e. parabolas, ellipses and hyperbolas, we have the 
exactly conesponding distinction drawn by Pappus (in. p. 54, 7 — 16) between 
plane and solid problems^ plane problems being those solved by means of 
straight lines and circumferences of circles, and solid problems those solved 
by means of one or more of the sections of the cone. But, whereas Proclus 



330 BOOK I [l3S 

and Eutocius speak of other solid loci besides conies, there is nothing in 
Pappus to support the wider application of the term. According to Pappus 
(ill. p. ^4, i6— ai) problems which could not be solved by means of straight 
lines, curdes, or conies were linear (yftofAfiuci) because they used for their 
construction lines having a more complicated and unnatural origin than thoae 
mentioned, namely such curves as quadrairias^ conchoids and dssoids. 
Similarly, in the passage supposed to be interpolated, linear loci are distin- 
guished as those which are neither straight lines nor drdes nor any of the 
conic sections (vii. p. 662, 13 — 15). Thus the classification given by Produs 
and Eutodus is less predse than that which we find in Piippus; and the 
indusion by Proclus of the cylindrical helix among solid loci, on the ground 
that it arises from a section of a solid figure, would seem to be, in any case^ 
due to some misapprehension. 

Comparing these passages and the hints in Pappus about loci on suffaoes 
(roiroi vpo« iwi^vtUf) with special refere